I'm wrapping my head around MLE right now and there's something about it that bothers me, irrationally I'm sure. I believe I understand the procedure: essentially we hold our observations fixed and maximize the likelihood function with respect to the parameters to find the parameters which would make a PDF that assigns a maximum value to our observations.
My question it this: why do we care about finding such a PDF? In particular, I'm imagining that we end up with a very skewed PDF so that the expected value is far from the maximum. Or what if we have an even weirder PDF than that? If $f$ is a PDF, it was my understanding that the number $f(x)$ is not particularly meaningful for continuous random variables—it's the area under the curve that we care about. So why aren't we in some way trying to maximize the area under our observations, or taking the expected value into account or something?
Hopefully this question makes a little bit of sense. I can try to clarify if it doesn't.