Three years after Muhammad (PBUH) was visited by Gabriel and received his first revelation from God, he started preaching revelations publicly, proclaiming that he is the last messenger of God, inviting people to Islam. My question is, how did he try to convince people that he is a prophet of God and people should believe in him? What reasons did he give for his claim?
Muhammad (SAW) was known by the community as a very kind and honest person who has never wronged anyone. This was even before the visit of Angel Gabriel.
People followed Islam and started to believe that Muhammad (SAW) because the Prophet treated everyone equally and fairly which meant better and happier life (especially for the slaves). He never showed anger and was always patient even when his enemies attacked him. He always offered peace.
He told the people about the message of Allah. The words of the Qur'an touch the hearts of many. Which lead to more and more people accepting Islam.
Just like any rassul, Muhammad (PBUH) too was gifted with miracles to convince people. However, the most important miracle is the Qur'an. At his time, there poetry competitions were very common, and when Muhammad (PBUH) brought some verses of the Qur'an, it leaved people speechless. You can read more about it in this article by Hamza Tzortzis.