Backward induction is how we usually solve sequential games (games where one player chooses first, and then the second player chooses, knowing what the first player has already chosen). It involves working out what the second player would do in response to each possible choice for the first player, and then using those outcomes to work out what is best for the first player to choose.
In the case of the pirate riddle, the process is somewhat similar. First, consider what will happen if there are only two pirates, then what will happen if there are three, then four, and finally five pirates. Pause the video before it gives you the answer, and see how you go. I bet you'll be surprised by what the Nash equilibrium is!
There are two problems with the pirate riddle game though. First, it assumes that the order is known. In reality, most pirates would elect the new captain once one was overthrown. So, the order of who would be the next captain is unknown and the game becomes more complex. Second, the pirate riddle that is presented in the video is a non-repeated game - that is, that the pirates only play this game once. In reality, the pirates will play the game many times - once for every time they capture some booty. If this game is a repeated game, and who would be the next captain is uncertain, then it is much more likely that pirates would share the treasure.
Believe it or not, there is actually an academic literature on the economics of pirates, and an excellent book by Peter Leeson (The Invisible Hook: The Hidden Economics of Pirates) summarises some of the key points of the literature (including how pirates divided their booty up, and how captains were selected). It is worth a read - you'd be surprised by how democratic (and rational) pirates were.