It really depends upon the Link in question.
In ALttP he probably did it for the good of his dearly departed Uncle who died trying to do the task Link was trying to do.
In LA, he did it simply because he had to do it in order to leave and get back to Hyrule.
In OoT, the same thing, doing it for the Deku Tree who died and in his last moments, entrusted him with the quest.
At first in MM, he was probably doing it in order to regain his original form, but was sucked into his quest by the plight of Tatl and Tael, whom he felt sorry for and then ultimately to protect the people whom he became fond of as well.
In WW, it was to save his sister and eventually, it became about saving the beautiful world he explored and saving Tetra, as well as something for his friend, the King of Red Lions.
In MC, it was to save his dear friend Princess Zelda from a fate as a statue for the rest of time and also, over time, to return Ezlo to his former self.
In TP, at first it was to save the kids and his supposed love interest (Ilia), but it soon became about doing it for Midna and Zelda.
In PH, it was to save Tetra from the horrors of the Ghost Ship and then return her to normal, as well as restoring Ceila's memory and helping Oshus.
In ST, it was to restore Zelda.
Or, more likely, it's simply because the player wants to play the game.