Wa Alaykum Assalam
This is a fascinating subject. I'm going to give a simple explanation for now, we must remember that every Prophet, from Adam (AS) to RasulAllah (SAW) were Muslims, and transmitted the message of Tawhid and Islam to their people. Islam was always there, God chose it as the religion of mankind from the creation of Adam, each and every Prophet was a Muslim - it just happened that as with Jesus (AS) or Moses (AS) for example the people corrupted their original message and scriptures of Tawhid over time and drifted away from the Truth.
With RasulAllah (SAW), he is the final messenger sent by Allah, Allah promised to protect this final revelation from corruption, which is why the Qur'an is today as it was since it first came, unchanged and true.