To be on one's last legs means to be worn out, tired, run down, and ready to die or otherwise cease working. Some examples I've found are
Grandfather is on his last legs. He'll be on his way to Heaven soon.
I just ran a mile to tell you this; I can't walk up the steps. I'm on my last legs.
My car is on its last legs. I doubt it will get me down the street to the used car dealer.
I've searched a bit on the interwebs, and while definitions abound, I can find no reference to the origin of the phrase. Why "last legs"? What happened to the first ones? What has interchangeable legs anyhow?