"We interrupt this program to bring you: Courage the Cowardly Dog show, starring Courage the Cowardly Dog! Abandoned as a pup, he was found by Muriel, who lives in the middle of Nowhere with her husband, Eustace Bagge. But creepy stuff happens in Nowhere... It's up to Courage to save his new home!"
That is... pretty much what you need to know in order to get the show. However! We get some clarification on that "abandoned as a pup" bit, late in the fourth season in the episode "Remembrance of Courage Past". Here's what happened: As a really young puppy, Courage was playing ball with his parents when circumstances led to him getting his head stuck in a metal gate. Unable to free him from it, his parents took him, gate and all, to the vet in order to get it removed. Unfortunately, this vet was running an experiment that involved breeding dogs in space, and he'd decided on Courage's parents becoming new test subjects. He took them both to a launch room in the back of the building and locked them both into a rocket. Courage, so young at the time, couldn't them in time, only being able to evade the vet until he found a garbage chute and escaped.
Ending up in the alley behind the building, all he could do was watch the rocket go up into space, taking his parents with it. Muriel found him there not too long after that, and decided to take him home with her. Thinking how brave he was to be in such all alone, she named him "Courage" right on the spot. From there on he lived with her and Eustace on that farm in the middle of Nowhere(, Kansas apparently), dealing with all sorts of inexplicable things that threatened his new home and owners.
Later on down the line, seeing a missing dog notice on the back of a milk carton sent Courage into a near catatonic stupor as he remembered the prior events. Out of concern, Muriel decided that the three of them would go see the vet. And of course it turned out to be the exact same guy from before. Freaking out, Courage quickly beat it and wound up at the same launch room before the vet caught and threw him into another rocket. Unfortunately, Muriel and Eustace witnessed this, prompting the vet to toss them into the rocket as well in order to silence them. Because of this though, Courage could slip out of the rocket just as he closed the rocket's door on them. And although Courage got caught again soon enough while trying to get the door open, he managed to distract the vet, get the key to the door, save Muriel and Eustace, and toss the vet inside all before the launch countdown hit 0.
With his family saved, and all (relatively) well again, they went back to their peaceful home and normal lives. Or at least as peaceful and normal as it ever got in Nowhere.