Disclaimer: I'm no expert on Islam. People like Sam Shamoun and David Wood are far and away more expert on Islam than I am.
I'm open to correction, but here's my understanding of the Satanic verses and the events around them:
1. When many of us Americans hear the phrase "the Satanic verses" we mostly associate it with Salman Rushdie's infamous satirical work.
2. However, the origins of the Satanic verses are far older than Rushdie. In fact, the Satanic verses, how they came to be, what they originally meant to Muhammad, etc., are found in the earliest historical accounts of Muhammad's life.
To be more specific, the Satanic verses are found in the biography of Muhammad called The Life of the Messenger of Allah by the Arab Muslim historian Ibn Ishaq (704-768 AD) as well as in the historical works of the Persian Muslim polymath Muhammad ibn Jarir al-Tabari (839-923 AD).
Muhammad himself lived from 570-632 AD.
I've only named two sources, but there are many other sources which describe and explain the Satanic verses (e.g. Ibn Sa'd al-Baghdadi, Muhammad al-Bukhari). Some three dozen sources or so. I presume the earlier sources are in general more reliable.
3. What are the Satanic verses? Relying on the earliest sources, the Satanic verses refer to words initially uttered and claimed by Muhammad to be divine revelation from Allah (via the angel Jibril aka Gabriel), but which Muhammad later renounced and instead claimed were words which Satan had tricked him to utter. The Satanic verses were originally thought be Allah's revelation to Muhammad in chapter 53 of the Qur'an.
Hence, according to Muhammad himself, the Satanic verses were not Allah's words, but in fact deceptive words spoken under Satanic influence.
4. Here are the Satanic verses in question:
Have you considered al-Lat and al-Uzza, and Manat, the third - the other one? These are the exalted gharaniq, whose intercession is hoped for.
The names "al-Lat," "al-Uzza," and "Manat" refer to three pagan goddesses which were commonly worshipped in the Mecca of Muhammad's earliest days, while "gharaniq" refers to the long-necked bird known as the crane. These "cranes" carried people's prayers to Allah. Thus these goddesses or "cranes" were regarded as intermediaries between the people and Allah.
In other words, according to the Satanic verses, Allah sanctions people praying to these goddesses as intermediaries. As such, people could continue to pray to these goddesses as they had been doing in the Mecca of Muhammad's day. It's basically a compromise position between pagan idolatry and full-blown submission to Allah. Allah the monotheistic "God" sanctions polytheism.
5. The Satanic verses were originally found in the 53rd chapter of the Qur'an (i.e. surah an-Najm). That is, the Satanic verses were in the Qur'an itself, at least for a brief spell.
6. However, today the Qur'an doesn't contain the second half of the Satanic verses. Instead, Qur'an 53:19-22 reads:
Have you considered al-Lat and al-Uzza, and Manat, the third - the other one?
These are the exalted gharaniq, whose intercession is hoped for.What! Are the males for you and the females for him [i.e. Allah]? Indeed that is an unfair division!
7. We have to look at the historical context in which Muhammad lived in Mecca in order to understand the reason(s) Muhammad is said to have thought the Satanic verses were divine revelation.
In Muhammad's day, Mecca was filled with the worship of various idols. Many merchants and traders made a lot of money in the worship of idols.
For years Muhammad had been preaching the monotheism of Allah with little success. In fact, he faced a lot of opposition from local pagans including violence. He was banned from Mecca.
Yet, according to Muslims, Muhammad loved his people and longed for them to turn away from idols and submit to the real "God," Allah. He longed that Allah would give him a revelation that he could take to his people so that they would turn to Allah. Thus, in a moment of weakness, Satan took advantage of Muhammad's love for his people to deceive him.
8. However, it's more reasonable to consider the Satanic verses were an accommodation to polytheism because helped limit the persecution of the locals against Muhammad. If I recall, it allowed Muhammad to return to Mecca too.
9. What's more, according to the collection of hadiths (i.e. the extra-Qur'anic sayings of Muhammad) called Sahih al-Bukhari, which are considered by Sunni Muslims to be authoritative: when Muhammad received the divine revelation of chapter 53 of the Qur'an from Allah (via Gabriel), Muhammad, his followers, and the local pagans (as well as the jinns) "prostrated" themselves.
If so, then this leads us to ask, why would pagans prostrate themselves to a revelation from Allah if the Satanic verses did not contain the accommodation to polytheism?
10. Given all this, Muhammad apparently couldn't differentiate revelation from Allah (via Gabriel) from Satan. He couldn't tell who was inspiring him. This undermines Muslim contentions about how Muhammad received the Qur'an or Allah's divine revelation. The divine revelation and transmission process from Allah to Muhammad may not have been so flawless if Satan could interpose his polytheistic words into Muhammad who wrote it into chapter 53 of the Qur'an.
11. Interestingly, Muhammad's very first ever encounter with the angel Gabriel to receive divine revelation caused Muhammad to think he had been possessed by an evil jinn. He said he felt as if he were being overpowered and even strangled by Gabriel. He was in tremendous fear.
In fact, he even wanted to commit suicide, for either he felt it'd be better to be dead than possessed, or that eventually he would be driven mad by the jinn to commit suicide anyhow. He wanted to throw himself off of a cliff.
It was mainly his wife, Khadija, who convinced Muhammad he was not possessed. Moreover, she convinced Muhammad he must have been experiencing revelations from the divine, not from an evil jinn. She convinced him he must be a prophet, not a demoniac.
One wonders if Muhammad's original impression about his experience with the alleged angel Gabriel wasn't the true one. And I could see a (pagan) wife who loves her (pagan) husband offering this alternative interpretation in order to keep her husband from killing himself.
12. According to Islam, shirk refers to the practice of polytheism. It is also an unforgivable sin in Islam.
However, if the events around Satanic verses are true, then Muhammad committed shirk.
13. Finally, for now, Islam claims the Bible itself verifies Islam. Yet, according to the Bible in Deut 18:20-22:
'But the prophet who presumes to speak a word in my name that I have not commanded him to speak, or who speaks in the name of other gods, that same prophet shall die.' And if you say in your heart, 'How may we know the word that the Lord has not spoken?' - when a prophet speaks in the name of the Lord, if the word does not come to pass or come true, that is a word that the Lord has not spoken; the prophet has spoken it presumptuously. You need not be afraid of him.
If the Satanic verses and the events around them are true, then Muhammad would have been a false prophet, and would have deserved to die, for he spoke in God's name, but then what he spoke turned out to have been inspired by Satan, not God. In addition, what Muhammad said in the Qur'an didn't come to pass or come to be true, but instead was "abrogated" (as Muslims might say).