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The rules are the same as in FIDE chess, with the following exceptions.

-You may capture your own pieces.

Why? 

I am trying to figure out the purpose of this game. 

Sacrifice a piece to develop another, move a knight to where you can capture it and promote your pawn- The sacrificing isn't a move that would be used extremely often, but it does add more strategy to trades and sacrifices.

What happens in mutual stalemate, where a player makes a move that leaves both sides unable to move?

You don't have to attack your own King for forcing checkmate with a Rook, do you?

It is not completely clear what "being in check by your own pieces" means. In FIDE rules the checking rule is formulated as that you cannot expose your King to (pseudo-legal) capture. But my own move would never do that for my own pieces, as after that it is not my turn to move anymore. And it is conceivable that the opponent is forced to resolve the self-check before it becomes your turn again, or that this turn will never come. E.g.

When white plays Qe2 here black is checkmated, and if that terminates the game there is no danger that the white King would be captured by its own Queen. Also, if black is on move here, can he play Ke2, because the Queen is not allowed to capture him, as this would also attack her own King? In FIDE rules capturing a King trumps the checking rule, and is allowed by any pseudo-legal move. But in Atomic Chess the rule that you cannot blow up your own King trumps capturing/blowing up the enemy King, and Ke2 would indeed be allowed in the given position.

So it is important to specify the priority of the rules, in particular what would happen in a position where you can both capture your own King and the enemy King. The opponent could put you in a discovered check from your own piece. Can he do that while exposing his own King to capture elsewhere, because you must save your own King from self-capture first? Would you already have lost, because you must capture your own King even when you can also capture the opponent's?

Could you give an example of a position where it could happen? And it would be considered a win, but I can't quite think of position where that could occur, since any stalemate-giving piece could simply be moved.

I cannot even imagine how it could happen in principle. You can only be stalemated if it is your turn.

Ah, I see the problem. But the main point of the self-check rule is to force unique strategy in positioning, so I understand what you mean and I will change the wording to clarify.

The author, Rose Thorpe, has updated this page.

Something like this? Noting that the bishop would threaten its own king (without checkmating the opponent) if it took the rook, neither side has any valid moves.

Though there seems little reason to judge that differently from a win for the last to move