For western history, the important development that was needed to believe in life on other planets (including the moon) was the realization that planets were made of the same materials as earth (rock, water, air, etc.)

In the leading medieval cosmology, everything beyond the earth was thought to be made of a fundamentally different kind of matter. They considered it this way: rock naturally falls towards the middle of the universe. Our earth is at the middle of the universe for this very reason. If other planets were made of rock, they would fall toward earth. But they are not falling toward us; they move in perfect circles around us. Therefore, they must be made of something completely different. Also, because everything beyond the earth is much more perfect (i.e. it moves in perfect circles for eternity), the stuff that they are made of must also be much more perfect. This perfect matter cannot possibly have flaws or defects, so it must necessarily be perfectly smooth.

This was basically the idea they had until Galileo and other astronomers developed a new system of cosmology in which the planets were made of rock and were full of mountains and seas. After that, the idea that these planets had life was quite obvious to most people; because why would earth be the only planet with life? Surely God had a reason to make this infinite amount of stars and planets, and surely he would not waste all this space in the universe without filling it with life.