Laws of physics are (almost) time symmetric, so a time-reversed description of a physical process is as qualified as the original one. What's the reason then, that in reality one version seems to prevail but not the other?
Entropy is smaller closer to the big bang and larger on the other side, so it's asymmetric with respect to time. But asymmetry is not direction. "Entropy increases from big bang forward" and the time-reversed version "Entropy decreases towards future big bang" are equally good descriptions of this asymmetry. Are we choosing the former over the latter? If so, mustn't there be some reasons other than reversible laws and boundary conditions that legitimize our choice? What might they be?
Everything we remember happened when entropy was lower, not when it is higher. Could this be one of the reasons why we choose one over the other?