I may "stop" reading a book, to do something else and continue reading later. If I "finish" reading a book, I have reached the end of the book. I expect that rain will stop, and may start again later, or tomorrow (or next year!).
"Stop" has a sense that it can start again while "finish" means that it has ended. I don't normally know that rain has definitely finished.
You can't say that in english. You would have to say "It stopped raining". "It stopped to rain" doesn't really make sense, but it kind of sounds like it would mean "It (Something) stopped [doing an action] [in order] to [begin] rain[ing]"
A better example that may make sense are the 2 sentences "He stopped running" and "He stopped to run". The first one means, He ceased the action of running. The second one means, He stopped doing some other action in order to begin running.
As a native english speaker, I can't exactly tell you why your answer doesn't make sense, but I do know that it doesn't mean what you had intended it to mean.
'Rain' are the droplets of water that fall from clouds when there is moisture in the air, 'Raining' is the action of those droplets falling. While a native English speaker would understand what you mean if you said 'It stopped to rain' we would not say it that way, we would instead say; 'it stopped/finished raining' or 'the rain has stopped/ended' meaning that the droplets stopped falling from the sky...
..."It started to rain" is an appropriate way to say that drops of water have begun falling from the sky, like saying 'the droplets started to fall' but just as we would not say 'it stopped to rain' we also would not say 'the droplets stopped to fall' instead we would say 'the droplets stopped falling'.
That, i think, would be more like saying "Det slutade att regna" which i assume is wrong in Swedish also. I was lookimg to see if there were any comments asking why "It's stopped raining" short for it has stopped raining is not accepted. I know it might not be a literal translation but at least where I'm from in England you're more likely to hear this version spoken. The given version would be used in a larger sentence : when it stopped raining they went out to play.
I assume this comment was meant as an answer to AlecHirsch1:s comment. If that's the case, no, It stopped to rain would not be like det slutade att regna, it would be like det stannade för att regna.
det slutade att regna isn't wrong per se, you can use an att with sluta if you want to. I'd say it sounds better without it, but it isn't wrong and it doesn't change the meaning.
The reason it has stopped raining is not accepted because that would be det har slutat regna in Swedish. We ask you to keep the same tense in translations unless it's necessary to change it for some grammatical reason. Det slutade regna would be used like It stopped raining, for instance in a story set in the past tense. It doesn't mean the same thing as It has stopped raining.