The reasoning is that in reality, the game is about who kills the other king first. If the black king takes the white queen, then the black king is killed by the white rook, ending the game. The black queen loses her crown before she can kill the white king.
Can someone explain how this works?

The reasoning is that in reality, the game is about who kills the other king first. If the black king takes the white queen, then the black king is killed by the white rook, ending the game. The black queen loses her crown before she can kill the white king.
Thank you for the explanation! This helps greatly.
After Qxf8+, why can I not take the queen back? My queen is pinning the rook to the King. Therefore, I should be able to take the queen right? As I found out today, this is not the case. So what is the reasoning behind this?
Jeremy nailed it. However, as of today's conventions, kings are not allowed to be captured. So, instead of allowing kings to be captured, we consider such a move as illegal - because capturing the queen would have allowed the Black king to be captured (obviously, like Jeremy mentioned, before the White king can be captured)

King's cannot move into check but if he has no legal move it is stalemate. If he is checked, the player must do all he can to get away from it. If he can't do it, he is checkmated.
Jeremy nailed it. However, as of today's conventions, kings are not allowed to be captured. So, instead of allowing kings to be captured, we consider such a move as illegal - because capturing the queen would have allowed the Black king to be captured (obviously, like Jeremy mentioned, before the White king can be captured)
If this is the reason behind it, then stalemate should not be a draw.
Had the king been allowed to be captured, stalemate would end in a win for the stalemating player since he has the option of capturing the opposing king regardless of the available king options for the stalemated side.
But as we know it today, illegal moves are forbidden from being played out.
That's why in stalemate, any move which the stalemated side would make is illegal. The stalemated side no longer has any legal move to make.
As to whether a stalemate is a draw or not due to "no legal move available", it's a separate matter altogether - but as we know it today, it's a draw.
On a separate and unrelated note from the OP's question, if the game could be played out to a king capture, there would be no such thing as stalemate, even in a king and pawn ending.
After Qxf8+, why can I not take the queen back? My queen is pinning the rook to the King. Therefore, I should be able to take the queen right? As I found out today, this is not the case. So what is the reasoning behind this?
So always remember - a pinned piece can still retain its ability of protecting a piece from capture by an opposing king.
After Qxf8+, why can I not take the queen back? My queen is pinning the rook to the King. Therefore, I should be able to take the queen right? As I found out today, this is not the case. So what is the reasoning behind this?