This game mode only appears when a timer runs to zero and there is no clear winner of a match. Those players tied for first lose their weapons and are put in their starting positions on the map with maxed-out damage and one Stock. The match ends when only one player remains, who is declared winner, or the last players are knocked out simultaneously, after which the match ends in a draw for those players.
When playing Sudden Death it is important to keep note that you can die in one heavy attack. The opponent knows this as well, so use that to your advantage, and while they are using their heavy attack get out of the way and then use a heavy attack on them while the attack animation is still going for them.