Nuke your opponent's king to win.
In addition to standard rules, all captures cause an "explosion" by which all surrounding pieces (not pawns!) on the adjacent squares (one square horizontal to, vertical to, and diagonally from the capture) and the capturing piece are removed from the board. Therefore, it is illegal to capture a piece that results in your king blowing up, nor can a king capture any piece (both of these actions would end his life).
A modification to the last line of FIDE rule 3.9 is made for Atomic. The original line says: "No piece can be moved that will either expose the king of the same colour to check or leave that king in check." An exception to this rule is made if you can blow up the opponent's king while in check. Simply put, any move that results in blowing up the opposite king will result in an immediate victory, overriding all checks.
A special note for Atomic games is that kings can be connected by moving one into the adjacent squares of the other sided king. When the kings are connected, checks do not apply. As it is illegal for a capture to blow up your own king, it is not possible to capture the other king directly. To "win" such an endgame, it would be necessary to cause the kings to detach by zugzwang, or that a piece of the opposite color is exploded while the king is next to it.
A checkmate is declared if a king is put in check, even if the king isn't immediately blown up, given that the king has no escape. The king itself also cannot capture any piece because that would result in a self-explosion, which is not allowed.
Find more explanations with examples on https://illion-atomic.netlify.com/atomic-rules/.