Firstly: Textual Evidence
The evidence includes the texts that came prophesying the advent of Islam and the Prophet of Islam, whether in the previous scriptures or in the Quran; the Book of Islam.
In Deuteronomy 18:18, we find that God spoke to Moses and said to him that He will raise from among their brethren a Prophet like unto him (Moses) and he (the prophet) will speak of God. In so many other places in the Bible, we will get the names of 'the Prophet' in combination with the messiah and the Baptist. This Prophet is none but Muhammad; the Prophet of Islam.
Also, he is referred to in the Bible as 'Elijah' coming from the Arabian Peninsula. There is also a reference to the light coming from the mountains of Pharan.
The textual evidence in the previous books are also referred to in the Quran in the stories of the Jews and Christians and the glad tidings given by Jesus to his people that there will be a final Prophet coming after him whose name will be Ahmad.
In Surah 3 (Aal-Imran), Almighty Allah tells us that He has taken the covenant from all prophets and messengers to believe in this final prophet who comes with the same message that is monotheism, the true message of Allah sent with all prophets and messengers.
Secondly: Logical Evidence
The very message of Jesus and his short stay on earth show that there was a need for another prophet to come settling all issues and showing complete course of guidance for all humans. This conforms with the saying of Jesus himself that he came not to initiate something new but to fulfill.
The world, before the advent of the Prophet of Islam was in a case of a total disruption and complete disorder. There were two main superpowers that engaged in a series of wars and fierce fighting, there was no place for the weak; rather, social injustice was spread everywhere and man’s condition went, in terms of morality, to alarming levels.
This unjust chaos and total disruption showed that there was a need for a religion that could re-establish order again. This was the condition of the world when Islam came with a complete social justice system, rights and equality, illumination and morality.
Thirdly: Other Evidences
There are so many other evidences that Islam is a religion of truth. Among them are the following:
The Quran is the eternal message of Islam and a timeless miracle that no one can produce the like thereof. Arabs, who were masters of linguistic excellence, could not produce the like of the shortest verse of the Quran when they were challenged to do so.
Islam calls for understanding and a deep search in one's self, in the creation of the heavens and earth and the contemplation in everything around man. Had Islam been a false religion, it would have asked man to cripple his mind and paralyze his intelligence and stop thinking about anything. (see Surahs 2:164, 6:11, and 6:95-100)
Modern science testifies to Islam; the Quran contains lots of scientific facts that were completely unknown at the time of its revelation. They were just discovered recently. This is in itself standing evidence on the truthfulness of the Quran. The Quran gives facts about the life of the fetus in its mother’s womb (23:12-14), the layers of the earth (76:2-3, 65:12), stars and planets, the fall of the rain and lots of other issues that are only known to scientists who are deep in knowledge.
There were lots of material miracles that were performed and eye-witnessed during the life-time of the Prophet Muhammad (peace and blessings be upon him). Among these miracles is the flow of water from between his fingers, the dividing of the moon into two pieces, his speech to a lizard which was able to tell that he is the messenger of Allah besides so many other miracles that were performed and eye-witnessed by his contemporaries.
These are but a few of the many evidences supporting Islam and showing that it is the eternal message of Allah and a religion of truth.