How do we know what we know?

[Angrboda]

This then begs the question: can we really know anything to be objectively true if everything we do passes through such imperfect filters as our senses?

It depends a lot on what you mean by knowing something to be objectively true. In a sense, science and empiricism are somewhat apophatic, a lot of it is concerned with determining what things are not as opposed to delineating specifically what things are (for example, Popper's rationalism, falsification, the null hypothesis and so on). What we typically refer to as objective knowledge rests on a number of assumptions, such as that the real world exists independent of our minds and that our senses and experience deliver valid information about the real world. It's impossible to know anything absolutely, and so our knowledge is framed within those assumptions. Within that frame, our senses, aided by methodological tools like science, we are able to produce models of reality which are useful and ostensibly true if our assumptions are true. Beyond that, I don't know what it would mean to have knowledge of objective reality.

You asked a followup question which, IIRC, was whether it was worth thinking about if the answer is no, that we can't have knowledge of objective reality. We don't actually have what classical philosophy would necessarily call knowledge of objective reality, but what we do have, given our assumptions, produces robust and reliable predictions about what we will experience if we do X or Y. That may not be absolute knowledge of objective reality, but since our experience is the only thing worth caring about, I hardly think the difference matters. Is it worth pondering the questions surrounding knowledge of objective reality? I think so, but then I'm somewhat biased as a philosopher with an interest in philosophy of science and epistemology. YMMV. But then, I only have to point to Karl Popper and Francis Bacon to vet my opinion on the value of thinking about such questions.