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I'm no expert on parliament or the Swedish systems. Here's the wiki: https://en.m.wikipedia.org/wiki/Riksdag
But for basics, based on my understanding:
A parliamentary system means the head of state/whoever has executive powers is chosen by the house, which in Sweden is one group of 349.
Now how members in that group are chosen can vary too. In Sweden, it's a proportional representation. So for simplicity, let's say it was 100 seats. After an election, the percentage each party won in votes represents how many seats they obtain (some countries will have a minimum percentage, or some other minimum barrier).
In Sweden, like many other parliamentary systems, there are multiple political parties who all own some chunk of the seats. Additionally in Sweden, no party by itself owns an absolute majority currently. So what happens to form a working government is parties form coalitions. Think of how in the US, a piece of legislation looks to win votes from members, sometimes even across party lines. The coalition in this case is a few different parties saying "look, we'll probably vote together on the same things to get stuff passed, so we'll form an official coalition". This allows the new coalition to govern, including stuff like choosing which PM goes up for vote.
(I'm using the party shorthand here, since otherwise it's long and confusing, especially for Americans).
Currently, the Riksdag is made up of 100 S, 70 M, 62 SD, 31 C, 27 V, 22 KD, 20 L, 16 MP, 1 Independent.
As you can see, no one has the 175 seats needed to be a majority.
So the coalition that happened waz
S, V, MP, C, Independent. That leads to 175 seats. S are the Social Democrats, who the above PM is from.
C stands for the center party. Meanwhile V and MP are further left parties. The S are generally in control, but need to cater to their coalition members, especially V and MP since they make up more votes.
So what happened here, from my understanding, is the center party didn't agree with the coalition budget proposal because it leaned too left, and they decided to not vote at all. Since all 174 of the opposition voted no, it led to the budget not passing.
Then the 174 of the opposition put to vote their own budget. The center chose to not vote again. And with that, the opposition budget passed.
The MP party quit the coalition in protest, basically. Since they quit the coalition, there is no majority. The PM decided to step down since technically she would be PM based on a coalition that doesn't exist anymore. That's why this all happened within hours.
It's not exactly that the government collapsed, but that the center didn't vote with their coalition members, and another part of the coalition left due to that. How the S party intends to handle governing while dealing with the center party, who knows. In the US, we have a similar situation where the ruling party cannot get enough votes to pass bills, the only difference is the opposition passing a bill won't happen due to veto powers and the current political make up.
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