But it's come into the Swedish language in a rather roundabout way.
In Danish and Norwegian, the verb 'to laugh' is le, which shares its linguistic roots with English 'laugh' and German lachen. These words are all related to the Old Norse word hlæja and, going even further back in time, Proto-Germanic klakhjan.
For many centuries, le meant 'to laugh' in Swedish as well, but as you may already know, today Swedish le means 'to smile', after skratta usurped it in its original use.
Skratta comes from an old Swedish word skrata, which meant something like 'to scare something away by making lots of noise', possibly related to an even older world used to refer to a mythological demon.
In Swedish, skratta at first meant something like 'to clamour' or 'to make a lot of noise' and then was used specifically to describe a particularly loud and emphatic laugh. Similarly, Norwegian has the verb skratte to describe loud laughter, while in Danish skratte means 'to rattle', all from the same root word. But from the mid-1500s, skratta began to be used in Sweden instead of le as the general term for laughter.
The reason skratta overtook le is probably a question of linguistic survival of the fittest. Because languages take in words from other origins all the time, there are often cases of two words fighting for usage. Sometimes a word might be more popular because it is seen as more elegant, because it fits into an existing linguistic pattern, or just because it's new, but sometimes it's a question of one word being linguistically stronger.
Le may have been too short and feeble compared to skratta, with its clashing consonants sounding closer to the act of laughter, and that's a likely explanation of how the meaning of le became diluted to a simple smile. This kind of language change doesn't usually happen consciously: it's not that Swedes debated the appropriateness of each word, just that more and more people gradually started using skratta.
There are lots of related words if you want to get specific: småskratta (literally 'small laugh': to laugh lightly or chuckle), gapskratta (to guffaw or laugh loudly), and hånskratta (to laugh mockingly, or to scoff). Also note the difference between skratta åt and skratta ut: both mean 'to laugh at' but the second phrase always has the sense of making fun of someone or something.
Det fick alla att skratta
It made everyone laugh
Du lovade att du inte skulle skratta!
You promised that you wouldn't laugh!