The Fermi Paradox is probably the biggest made up paradox in existence that still gets attention from retarded teens incapable of imagining the infinite(or nigh infinite) vastness of space and time in comparison to the time radios where invented to now. Also, radio waves get more indistinguishable from background noise the further they travel. Even so, if you were to map the waves sent from our system on the galaxy it would be a dot on the map. These people have never heard of Occam's razor, because the simple answer is just boring and can't be exploited.
It isn't even logically speaking a paradox. There is no contradiction between "aliens might exist" and "they are not here"

What's worse is that Enrico Fermi never wrote a single thing about extraterrestrials. He made an off-hand remark about some cartoon in the newspaper which turned into a discussion about the infeasibility of interstellar travel (this was before the moon landing)

Often times it's used as scarefuel by people who think that if we all don't go full space faggot communism with twitch streaming characteristics we're literally all going to die and it's the racist corporatist fascists' fault.
This "paradox" is so gay that it even has a tranny video game named after it.