Why do people believe in aliens?

Because they either want to, or have evidence and/or reason to believe so.

'Want' being from the fascination of extraterrestrials and their capabilities, and perhaps the personal attachment of being part of something more vast and mysterious.

'Evidence' being from the sightings, reports and documentations of over thirty million people, or perhaps personal experience only explained through the intervention of aliens.

And,

'Reason' from the ancient scientific and philosophical questions regarding the size of the universe and potential habitation of other life as evidence of our own existence.