The "Satanic Verses" refers to an incident in which Muhammad, according to Islam's own sources, mistakenly recited revelations from Satan.
The source we will look at for the Satanic verses is The History of al-Tabari (915 AD), which was written by the same author that produced the most important Tafsir (Quranic commentary) in Islam:
The History of al-Tabari, Vol 6, pg. 108-110
When the Messenger of God saw how his tribe turned their backs on him and was grieved to see them shunning the message he had brought to them from God, he longed in his soul that something would come to him from God which would reconcile him with his tribe. With his love for his tribe and his eagerness for their welfare it would have delighted him if some of the difficulties which they made for him could have been smoothed out, and he debated with himself and fervently desired such an outcome.
Then God revealed: By the Star when it sets, your comrade does not err, nor is he deceived; nor does he speak out of (his own) desire ... and when he came to the words: Have you thought upon al-Lat and al-'Uzza and Manat, the third, the other? Satan cast on his tongue, because of his inner debates and what he desired to bring to his people, the words: These are the high-flying cranes; verily their intercession is accepted with approval.
When Quraysh heard this, they rejoiced and were happy and delighted at the way in which he spoke of their gods, and they listened to him, while the Muslims, having complete trust in their Prophet in respect of the messages which he brought from God, did not suspect him of error, illusion, or mistake. When he came to the prostration, having completed the surah, he prostrated himself and the Muslims did likewise, following their Prophet, trusting in the message which he had brought and following his example.
Those polytheists of the Quraysh and others who were in the mosque likewise prostrated themselves because of the reference to their gods which they had heard, so that there was no one in the mosque, believer or unbeliever, who did not prostrate himself. The one exception was al-Walid b. al-Mughirah, who was a very old man and could not prostrate himself; but he took a handful of soil from the valley in his hand and bowed over that. Then they all dispersed from the mosque.
The Quraysh left delighted by the mention of their gods which they had heard, saying, "Muhammad has mentioned our gods in the most favorable way possible, stating in his recitation that they are the high-flying cranes and that their intercession is received with approval." The news of this prostration reached those of the Messenger of God's Companions who were in Abyssinia and people said, "The Quraysh have accepted Islam." Some rose up to return, while others remained behind.
Then Gabriel came to the Messenger of God and said, "Muhammad, what have you done? You have recited to the people that which I did not bring to you from God, and you have said that which was not said to you." Then the Messenger of God was much grieved and feared God greatly, but God sent down a revelation to him, for He was merciful to him, consoling him and making the matter light for him, informing him that there had never been a prophet or a messenger before him who desired as he desired and wished as he wished but that Satan had cast words into his recitation, as he had cast words on Muhammad's tongue. Then God cancelled what Satan had thus cast, and established his verses by telling him that he was like other prophets and messengers, and revealed:
"Never did we send a messenger or a prophet before you but that when he recited (the Message) Satan cast words into his recitation (umniyyah). God abrogates what Satan casts. Then God established his verses. God is knower, wise."
Thus God removed the sorrow from his Messenger, reassured him about that which he had feared and cancelled the words which Satan had cast on his tongue, that their gods were the highflying cranes whose intercession was accepted with approval. He now revealed, following the mention of "al-Lat, al-'Uzza and Manat, the third, the other," the words: Are yours the males and his the females? That indeed were an unfair division! They are but names which you have named, you and your fathers ... to the words: to whom he wills and accepts.' This means, how can the intercession of their gods avail with God?
When Muhammad brought a revelation from God cancelling what Satan had cast on the tongue of His Prophet, the Quraysh said, "Muhammad has repented of what he said concerning the position of your gods with God, and has altered it and brought something else." Those two phrases which Satan had cast on the tongue of the Messenger of God were in the mouth of every polytheists, and they became even more ill-disposed and more violent in their persecution of those of them who had accepted Islam and followed the Messenger of God.
The account is repeated again just a page later in volume 6, in which it is recorded that Muhammad said the following when Gabriel first informed him that he had recited from Satan:
The History of al-Tabari, Vol 6, pg. 111
That evening Gabriel came to him and reviewed the surah with him, and when he reached the two phrases which Satan had cast upon his tongue he said, "I did not bring you these two." Then the Messenger of God said, "I have fabricated things against God, and have imputed to Him words which He has not spoken."
In reviewing the whole account, we see that:
This account, then, gives us the background of two passages from the Quran.
The first passage is the one which originally contained the Satanic verses, from Surah 53, which now reads:
Surah 53:19-22
19 Now, have you considered ˹the idols of˺ Lat and Uzza,
20 and the third one, Manat, as well?
21 Do you ˹prefer to˺ have sons while ˹you attribute˺ to Him daughters?
22 Then this is ˹truly˺ a biased distribution!
The second passage is the one which Gabriel used to comfort Muhammad, in which he tells Muhammad that Satan caused all the prophets to do something similar to what Muhammad had just done:
Surah 22:52
52 Never did We send a Messenger or a Prophet before you, but; when he did recite the revelation or narrated or spoke, Shaitan (Satan) threw (some falsehood) in it. But Allah abolishes that which Shaitan (Satan) throws in. Then Allah establishes His Revelations. And Allah is All-Knower, All-Wise:
Therefore, it was Muhammad himself who claimed that he recited verses that were given to him by Satan, because he wanted to please the Pagan Quraysh, and this fact is attested to in the final text of the Quran.
The Satanic Verses show us that Muslims are entrusting their eternal destinies to a man who, according to his own testimony, could not even tell the difference between the voice of Satan and the voice of God. They have their faith in a man who was a Satanic false prophet, according to himself.
The Satanic Verses also show us that Muhammad was not a Biblical prophet, as the Bible says the following:
Deuteronomy 18:20
20 But the prophet, which shall presume to speak a word in my name, which I have not commanded him to speak, or that shall speak in the name of other gods, even that prophet shall die.
Muhammad, had he been alive at the time of Moses and Joshua, would have been put to death for the Satanic Verses incident, on the basis of his own testimony. Yet, what did Muhammad say in the Quran? "All God's prophets have had this happen to them". This totally contradicts the above verse of Scripture, which says that anyone who speaks in the name of other gods should be immediately put to death, not continue their career as a "prophet".
So, not only is Muhammad totally disqualified as a prophet, Biblically speaking, he also slanders all of the prophets in the Bible, which he claimed were his predecessors, by saying that they all spoke Satanic Verses, when in reality, according to Scripture, none of them ever did so, and if they had, they would've been killed.
In regard to Muslim sources for the Satanic Verses, al-Tabari is far from the only witness we have.
A nearly identical account to the one from al-Tabari above is also given in Sirat Rasul Allah ("The Life of the Prophet"), which is the earliest biography of Muhammad, written by Ibn Ishaq (704-767) and Ibn Hisham (died 833), under the section entitled "His Temporary Concession to Polytheism". In total, there are dozens of Muslim sources from the first few centuries of Islam that reference the event in some form - Shahab Ahmed catalogues 50 in his book Before Orthodoxy.
These accounts are also significant given the fact that, obviously, no one who is Muslim would have fabricated this story. Those who have read the Hadith know that you would have been killed for saying anything remotely like this about Muhammad amongst the early Muslims, if it weren't true (Sahih al-Bukhari 2934, 3031-32, 4037, Sunan abu-Dawud 4361-62, etc.). The only explanation as to why this is in so many early Muslim sources is because it actually happened.
So, what do Muslims do about this? Doesn't it present a big problem for them and their religion, which is supposedly the purest form of Monotheism?
No, because the Muslim response is to simply invalidate their own sources - the same sources which they use frequently for other purposes, and acknowledge as reliable as pertaining to many other topics, except this one. The way that they deal with this account, then, is by pretending it never happened, and not actually dealing with it at all.
On what basis do they invalidate these portions of their own sources? On the basis that they don't like them. It's as simple as that. That's the only reason you need when you belong to a false religion, and don't care at all about finding absolute truth.
The Satanic Verses weren't from Satan at all. They came from Muhammad, because he wanted his tribe to accept him as a prophet. Later, already being the false prophet that he was, he claimed a supernatural reason for his compromise.
In reality, the entire Quran can be classified as Satanic Verses. They were all given by a false prophet, who even if he did have encounters with an angel, that angel was Satan. However, given how horrible the Quran is, I am more inclined to believe that Muhammad made the vast majority of it up, rather than having gotten it from even a demon.