Why did Constantinople finally fall?

There are a number of reasons; however, the main ones are as below:

1- Byzantine Empire's army were outnumbered by Mehmet II's massive troops

2- Byzantine Empire had already been reduced to a city-state inside its strong defensive fortification, so Constantinople was an open target with no support from outside the city

3- The reinforcement from Rome, despite emperor's requests, was delayed (according to some intentionally), and

4- Mehmet II's army finally managed to open big holes in the city walls with massive cannons built by a Hungarian engineer, who had offered his invention to the Byzantine emperor and the emperor could not afford it.