In Mystery Science Theater 3000 the host is living on The Satellite of Love. What is the purpose of this satellite/spacecraft?
It is explained in the theme song:
If you're wondering how he eats and breathes
And other science facts (LA LA LA)
Just repeat to yourself it's just a show
I should really just relax
Don't try to take the show too seriously. It is a plot vehicle for having him be isolated with robots and watching bad movies.
At least one of the theme songs suggest that the host's presence there is because "his boss didn't like him", and another that the host is an experimental subject of some kind. We're told that the movies are out of his control.
The purpose may be as a isolation laboratory.
That said, it's not clear that the back story is either well thought out or in the least bit consistent.
The in-universe explanation is fairly simple: it's a movie theater in space, and there are three given explanations throughout the show for why it exists:
In the unbroadcast demo of the show, K00- The Green Slime which was first shown at a convention in 2008, Joel Hodgson (Changed to Robinson in Season 1) says:
My name is Joel Hodgson and this is The Satelite of Love, a space station of my own design, isn't she something?
Later though, the Satellite of Love is definitely for showing movies to a test subject, with various motivations:
Joel: Happy Near Years, Doctor
Dr. Forrester: Don't happy new year me, you white piece of trash floating in the vacuum of space, we just heard that the Russians have launched their own comedian into space and he's already pulling a 4 rating.
Dr. Erhardt: He's a regular Gallagher too, it's called Very Incredible Movie Theater 4.
Joel: Dr. Erhardt and Dr. Forrester, would it be too much to ask if you could let me and my friends know when we might be getting out of space?
Dr. Forrester: Sure Joely-poley, we're planning a show for you right now here on Earth.
Dr. Forrester: Yeah booby, it's about a guy and three robots. And they're submerged deep in the Atlantic trench, three miles under the ocean's surface and we send him transmission after transmission of Jacques Cousteau movies.
And again in K08- Gamera Vs. Guiron:
Dr. Forrester: Now listen up Joely boy, if you get any more ratings like last week, you're gonna be in trouble, remember: your rocket engines are controlled by ratings, if you don't shape up you're gonna find yourself swimming in the Indian Ocean without a paddle
Though it's hard to tell whether this remains the case (it's implied due to commercial sign, but that could just be breaking the fourth wall).
Dr. Forrester's opening monologue in Mystery Science Theater 3000: The Movie explains his motivation for the later series quite thoroughly:
Ah! You're here. Hello. Welcome. I'm Dr. Clayton Forrester, and soon you will all bow down before me. What you're about to see is an experiment and by observing, you have become part of that experiment. For I have shot a man into space and I'm driving him crazy by forcing him to watch the worst movies ever made.
Oh, sure, we all have thought about it, but I had the guts to do it!
This is my test subject: Mike Nelson. A disgustingly mild-mannered dope who has managed to survive every film I subjected him to. But perhaps this movie will drive him to the breaking point and crush his soul, and then I'll unleash it on an unwitting public, and then I will rule the world!
Then in the 500 years which pass between S07E06- Laserblast and S08E01- Revenge of the Creature, the human race goes extinct and Primates have taken over the planet (due to Mike's descendants exclusively marrying Apes). It turns out it's Ape Law that movies must be watched, as commanded by the Lawgiver. The final segment of the episode reveals that the Lawgiver is Pearl Forrester, Clayton's mother, who then explains her motivation to continue the experiments:
Pearl: Here's the short story: You see, after my beloved Clayton turned into a Star baby, I vowed to do it different. I vowed to raise him to be as fine a young man as you could ever ever hope to see...
But I got busy and I never got around to it, so I got another hopeless idiot who started off on his mad experiments....
And then it happened... my beloved Clayton passed away. When I was done smothering him I vowed to avenge his death, he asked me on his death bed to keep the experiment going no matter what the cost.
Tom Servo: This is the short story, right?
Pearl: So when I died, I asked someone to freeze my body, which they did and it was really quite reasonable. 500 years later the apes thawed me out, cured my freezer burn and that's why I'm the Lawgiver.
Tom Servo: Well I can see that's certainly makes a lotta HUH?!
Pearl: And I will keep my word, Nelstone, I will avenge my son's death. I will dog you to the end of the universe! [Manic laughter]
In S09E01- The Projected Man they manage to return to Earth through a wormhole and Pearl sets up in the ancient Castle Forrester, home of her ancestors, who it seems have always been performing these kinds of experiments:
Pearl: I found the ancient family diary of the Forrester clan, it's fascinating! [...]
Apparently my ancestors have always been involved in odd experiments; Amethystro Gento Forrester trapped a man in a cave and pushed in bad paintings of the hunt, Emerald Montgomery Forrester trapped a man on an island and forced him to read poorly-done parchments. And, could this be? Aventurine Salbeaudefor Forrester trapped a man in a tree and told him really bad sagas!
I have a feeling I'm onto something here Nelson, some power, some force beyond my control and it doesn't look good for you!
Later in the same episode we hear of her epiphany:
Pearl: Mike, I have reclaimed my destiny. As a Forrester I owe it to the world to take over the world! I shall dedicate my considerable genius to this and nothing else!..
Well I also do like those bus trips down to the dog track outside Omaha, it's a fun time and a real good deal and to tell you the truth I usually come out pretty good, I just have a feel for playing the puppies I just don't know what
COWER WORLD! TREMBLE WORLD! RUN AND MEWL AND PUKE IN TERROR WORLD!
And for the last two seasons she is driven by the same desire to control the world that her children (well, Clayton twice) and ancestors did too.