It is simply asking basically "if the universe is teeming with life, where is it?" and by that raising both the question of whether or not the universe actually is teeming with life, and if it is, why aren't they here? It reduces to an actual paradox only if you assume an answer to one part (the amount of life) that isn't reasonably compatible with the second part (why we haven't seen evidence of it) without a reasonable explanation.
That it simply on average takes too long for us to have seen evidence yet is in fact one of the many suggested solutions.
But overall the point of the Fermi paradox is that the problem if why we don't see aliens is two-pronged: They might never develop, or just not have developed yet, to a sufficient stage. But it's also possible they do, but that something else prevents them from contacting us.
Thus the idea of "great filters" for example. E.g. maybe the universe is full of life, but it's exceedingly rare for it to survive long enough to be able to make contact without being destroyed. Maybe the odds of getting killed off by asteroids is too high. Maybe war takes out most that get to a certain level. Maybe someone out there takes pot-shots at anyone sticking their head out? Maybe some galactic events sterilises large swathes of space too often? (intense gamma bursts have been raised as one possibility)
As well as many other options. E.g. maybe we're quarantined? Maybe we're in a simulation, and simulating aliens just isn't part of the program?