Well I haven't done any OCs yet but I'm in the planning of making one.
What inspires your costumes? Do you base them on a series or fandom, or are they completely your own creation?
My designs are based on characters in the various stories I write which have their own universes and such. Designing the outfits they wear is the hard part as my characters usually tend to change clothing on a regular basis with only keeping a few pieces similar between them.
Do you give your characters a backstory, or come up with "in-character" explanations for parts of the costume?
Yup. I tend to make sure all characters have a long, thought out back story that explains the reasons they are the way they are.
Overall, why do you do it? Do you like the freedom of original characters/designs? Do you like the challenge of coming up with a design from scratch, or do you find it more personal than strict recreation?
I also agree with shinseikakumei with it being personal. One of my characters, I had for over thirteen years now and she went through major developments over the years. I'm finally starting to seriously think about cosplaying her now that I have more skills since her sword is complicated.