What kind of timer socket is this and how can I get it working?
This is a GE device, though I think a Leutron or Leviton also make a pretty similar unit.  Here's a link to it on Home Depot's website.

My experience with these units is that they fail very rapidly - I went through 3 of them in a year before I switched brands.  You can see the comments on Home Depot echo this. They also can fail closed so that with a dead switch, everything is still powered and you have no way of turning it off - quite annoying if you ask me!

I would recommend that you replace it with another brand. They are not serviceable.