Why don't anime shows finish the manga? On almost every anime I've watched, they stop producing the anime even though there is more story in the manga, and then to get the rest of the story you have to read the manga.

For example, I've watched Maid-Sama, Fruits Basket, Ouran High School Host Club, Magi and none of them finish off what they started. The only show I've seen that has tried to animate the whole manga so far is Fairy Tail.

Why don't producers just animate the entire manga, or why do they start animating the manga and never finish? Why don't they finish the anime based on the manga?
As some of the comments have pointed out, anime is expensive to make and is often just advertising for the manga and merchandise. Another important aspect is that at the start of a season, the producers have to decide how much of the material they're going to adapt. They may be limited by whether the manga has already finished or not, or whether it has a good existing stopping point that they can use for a season finale.

Given a standard 13-episode season, it makes sense to try to pace the story out so that it covers a decent amount of ground before hitting some kind of climax in the finale (compare that to the evergreen series that can have 30 consecutive filler episodes before randomly having a massive battle scene and then continuing the story like nothing happened). So if there is enough source material, the producers may have a bit of a choice at the start - they can either try to cover the length of the material by skipping to the good bits, making for a solid show with a convincing ending but where things aren't too fleshed out. Or, they can take the first 1/2 or 1/3 of the manga, follow the story more closely, work in some of the more interesting small details, but finish on a weaker note.

So, I guess your question is, why go with the second option most of the time? And the answer is that by doing so, they keep the possibility of a second season open. The hope is always that they will get to keep going and finish the story. After all, twice the seasons means twice the profits, twice the audience who can then go and buy merch, and twice the amount of material that can be turned into merch (well it doesn't mean any of that but it sounds nice in theory). Except, as it so happens, for whatever reason these shows don't rate so well and hence they get dropped and the producers move on to the next shiny new thing.