To understand what happens when there are 5, 15, or even 150 pirates on the boat, it will help us to first consider the case when the lowly deck swabbing Pirate 1 is the only one left on board and suddenly finds himself as the most senior pirate aboard.
In this situation, when there is only one pirate left on the boat, Pirate 1, can vote for his own proposal and get 100% of the vote; so, he would simply take all of the gold.
If there were two pirates left, Pirate 2, now being the most senior pirate, could propose any plan since his own vote guarantees 50% of the total votes. Being the greedy gold-loving pirate he is, Pirate 2 would take all of the gold and Pirate 1 would be left with nothing.
If there were three pirates, Pirate 3 needs one other person to vote for his plan. The trick to this puzzle is understanding that if Pirate 3's plan is voted down, he would die and then there would be only two pirates on the boat. We already figured out what happens when there are only two pirates on the boat. In the case of two pirates, Pirate 1 receives nothing, so Pirate three can simply offer Pirate 1 a single gold coin and ensure his vote. As a perfectly rational pirate knows, one coin is better than no coins at all!
If there were four pirates, Pirate 4 needs to convince one other person to guarantee 50% of the vote. He could give Pirate 1 two gold coins, but his greed makes him realize that if his plan is scuttled, there will only be three pirates on the boat. When there are three pirates left, Pirate 2 knows he will get nothing; so Pirate 4 buys Pirate 2's vote with one gold coin.
Finally, when there are five pirates, Pirate 5 needs two other cohorts. He realizes that if he dies, Pirate 1 and Pirate 3 will get zero gold. So he offers each of them one doubloon and makes off with the other 98 pieces o' eight.
The pattern should be evident now. When 15 pirates are on board, Pirate 15 needs 7 other people to vote for him. He recruits pirates 13, 11, 9, 7, 5, 3, and 1 with one gold coin each, leaving 93 gold coins for Pirate 15. Those pirates will all vote for Pirate 15's plan because if they don't, they'll be stuck with Pirate 14's plan in which they all get nothing.