I don't particularly care about Open Source, but I do care about Free (as in Libre) software. I use it so I as a user can decide what it does, not a company or the developer. The four freedoms: to use, study, share and improve the software are the ones I am interested in when I use software.
Because I like having the ability to see and modify what it’s doing. Sure, I probably won’t touch the source of large GUI programs like Firefox, LibreOffice, GIMP or Inkscape, but having the possibility is comforting.