To see into the problem, consider the last case first. If there are only two pirates left, the Lieutenant will propose that he keep everything, and the Ensign’s protest will be outvoted. At least he doesn’t get killed.

If there are three pirates, the Lieutenant has an incentive to vote against whatever the Commander proposes, because if his plan is turned down, the Commander will get killed, and then the Lieutenant will get everything. However, and here is the insight, the Commander may be able to win the Ensign’s vote by offering him a coin. Nothing for the Lieutenant because his vote isn’t necessary. By giving up one coin, the Commander gets everything else. 

Consider the four-pirate case: As in the three-pirate case, the Captain needs only one vote other than his own to prevail, and he can get it by bribing the Ensign. The Commander has no reason to throw in; he will get his own deal in a few moments and he will prevail by bribing the Ensign. The Lieutenant might sell his vote. If it gets to a three-pirate situation the Lieutenant will get nothing, and it will never get to a two-pirate situation, so for him it’s now or never.  But the Ensign will certainly sell his vote.  So in a four-pirate case the Captain can prevail by buying at least one vote from either the Commander or the Ensign.

Now the five-pirate problem. The strategy is clear, you must buy the votes of at least two other pirates by offering them more than the Captain or Commander are likely to offer. The Captain will not sell his vote because if you are defeated he will have his own chance. The other three are in a position to bargain with you, and if they cooperate they can force your bid up. But the chances are very good that you can win the votes you need by offering to give any two of them a couple of coins. It is a strategy as old as King Edward I of England, who enlisted the support of his people against the rebelious Barons. And as fresh as "populist" presidential candidates who promise to favor the common man.

The great insight of game theory is that most games in the real world are played over and over again until you reach an equilibrium of interest and reward.  What proposal would you make if all the other pirates had studied the problem carefully, and even played it out a few times?