Why is it traditional to remove your hat out of respect, especially in the context of mourning?

I think this question may suffer from being a bit too broad. The tradition you’re mentioning isn’t present everywhere, and not even is it universal in Western culture. The practice in some situations is tied to religious significance which non-observers are not obligated to adhere to, e.g. in many sects of Judaism:

Not only do the Jews keep their hats on their heads at funerals, but in some countries they still wear black caps at weddings, in token of mourning for the destruction of the Temple. (Wagner, Manners, Customs and Observances)

But limiting ourselves to a general overview of the Anglo-Western tradition, as a "manner" or expression of civility, Brewer's Dictionary of Phrase and Fable declares removing one’s hat to be:

a relic of the ancient custom of taking off the helmet when no danger is nigh. A man takes off his hat to show he dares stand unarmed in your presence.

This is a rather general statement, unfortunately, but may speak to the notion of removing one’s hat at the dinner table or even just in one’s home being a sign of comfort - specifically, that the owner of the home has provided a place of sufficient refuge. The consensus seems to be that hat removal customs translate into our modern practices via the Victorians (unsurprisingly). Dress for Deference and Dissent: Hats and the Decline of Hat Honour (Corfield, Costume 1/1989; 23(1):64-79), traces the evolution of especially British hat decorum, arguing:

Using an everyday and highly visible item like a hat was an effective and very personal means of communication.

And related to our conversation here:

during the eighteenth century, it became decreasingly common for men to wear their headgear - other than wigs and nightcaps - indoors, as social etiquette became increasingly formalised and as housing also became better heated. That applied both in private homes and at public assemblies unless the purpose of a gathering was military or ceremonial.

And:

Above all, as the head was symbol of authority, the covering or uncovering of the head, in Western society, was for men an important signal of relative status. Whether and how universally regarded in practice is impossible to ascertain; but the expectation was clear enough. The traditional custom was expressed in the Princeton College rules of 1756. They declared that: ‘Inferiors, when they come into the company of a superior or speak to him, shall show their respect by pulling the hats’.

It’s clear that socially-conscious people of this period studied hat etiquette as a means to indicate, refine, and/or advance their social standing, so this could be the mechanism by which these behaviors were passed on. But as Corfield points out, regardless of evolving expectations on hat wearing in the public and private spheres, a constant was that it remained a gesture of respect to remove it, especially at funerals. Perhaps this was because the funeral was, generally, inherently non-secular and constant, and therefore not subject to the whims of fashion that were readily participated in outside of this context. Both the Catholic Church and the Church of England required parishioners to remove their hats during worship and, by extension, religious mourning.

EDIT: Clarity.