The way they get to the other convictions (I think) is a bit nuanced.Maybe on the 2nd degree manslaughter. The prosecution has already lost the case on the first two charges if we're all being honest.
The premise is that an officer can commit an assault while effectuating an arrest. The only reason officers are allowed to lay hands on someone is because they are arresting someone, and they are allowed to use a certain amount of force to do that. That is, if the force used is beyond that which is necessary (or mandated by policy), then it converts to a crime: battery (or assault or whatever they have up there). Basically, the officer would commit a separate crime by being too violent.
If you or I lay hands on someone, without intending to kill them, but they end up dead, then that's enough for a criminal conviction. Even if it is in self defense, in the scenario where the initial use of SD might be justified, but we carry it too far. Someone comes at us with a knife, we disarm them, they run away and we chase them, tackle them, then stab them. If they end up dead, we probably committed a crime.
I think that's the logic the prosecution is going for. Chauvin may have been justified initially, but it crossed over at some point into something criminal.