I get asked this question a lot, and am not very happy with any of the answers.
Vaguely I think of subfactor theory as a generalization of representation theory of groups. That is, if you have a group/subgroup subfactor, and look at the fusion category, you get a category that has all of the induction and restriction data for the subgroup (I think). So maybe that is my first question, what group theoretic information can you extract from the fusion category of a subgroup subfactor? You can do a similar thing with representations of quantum groups, and you also get "sporadic" subfactors, which don't come from groups or quantum groups. This is interesting because it looks a little like the classification of finite groups, you have a bunch of families and then the fun unexpected sporadic groups. I would love to hear someone else's opinion, if anyone can make the subfactors/group theory analogy more formal.
The other explanation as to the interest in subfactors that I hear is, "they have a lot of surprising connections to other topics and show up all over the place." Can someone tell me about such an instance? What are some other topics where subfactors unexpectedly showed up? I am vaguely aware of something to do with random matrices ...
Of course, the easy answer is that they are really beautiful and cool in their own right, and no one has to convince me of that.