Using 'even' this way is an intensifier, and is fairly modern, as well as being very conversational, so that it may be hard to find in dictionaries.
'Why am I even listening to you?' means the same as 'Why am I listening to you?' but with more force.
'What does this even mean?' means the same as ' What does this mean?', but, possibly, with the implication 'I really don't know what this means, and I suspect that not many others do either'.
In my opinion, there may be sometimes be an element of frustration being expressed also.
I have found a University of Pennsylvania language blog where the author says it seems to be 'purely emphatic'. He also wrote (in 2011) that he could not find any examples older than 'around 15 years ago' (so around the mid 1990s).
The blog author writes:
But there's a class of (I think) recent uses of even that resist this
sort of treatment as "intimating … an extreme case of a more general
proposition", and instead seem to be purely emphatic. I associate
this with the speech of young women, as in this passage from Rhoda
Hayter, The Witchy Worries of Abbie Adams (2010):
How'd you like to be innocently walking in the door one day with an
armload of books for your kitten and hear this in your head: "Father's
going to owe me a lot of dimes soon"? How random and creepy is that
???? What does something like that even mean???
The author goes on to say that the generalisation that this type of speech is mainly used by 'young women' is inaccurate.
Later, he gives some examples:
How do you even spell that?
How do you even pronounce Ophiuchus?
Why are you even bothering me?
If she has children then it will be because she wanted to and I don't see how that's even entertaining to anyone.
I have no idea who that even is.
but has anyone even considered how that even happened?
I can't even stand this thing anymore.
As far as I can tell, this purely-emphatic version of even is a
polarity item, in that it only works in a range of contexts including
negation and questions.
What does "even" even mean? Mark Liberman (Dept of Linguistics University of Pennsylvania)