Why do mitochondria fuse together?
Mitochondria produce a large quantity of reactive oxygen species (ROS) that damage the mitochondrial genetic material. These cell organelles do have a  DNA repair mechanism similar to that of the nucleus to some extent 3 which is not entirely understood however they have multiple copies and so the fusion and fission of mitochondria work together to maintain as many viable/healthy mitochondria as possible.

When two mitochondria fuse, their genetic material can recombine and thus exchange their damaged DNA. In this way, one of them repairs its DNA at the expense of the other accumulating more damaged DNA. Mitochondria that accumulate too much damaged DNA will 'die' - get eliminated, but they can be replaced by fission of healthy ones. Take a look at this wiki page, it has useful info.

Edit: This article may also be useful. (Mitochondrial fusion and division: regulation and role in cell viability
Giovanni BENARD and Mariusz KARBOWSKI, Semin Cell Dev Biol. 2009 May;20(3):365-74.)