But how can the judge do that when Dominion did not have to prove anything?

But they did have to prove things. This lawsuit has been going on for two years. Dominion got access to the Fox emails and texts in discovery, they deposed witnesses. They turned over their source code to Fox. Page 43 the judge says

While the Court must view the record in the light most favorable to Fox, the record does not show a genuine issue of material fact as to falsity. Through its extensive proof, Dominion has met its burden of showing there is no genuine issue of material fact as to falsity. Fox therefore had the burden to show an issue of material fact existed in turn. Fox failed to meet its burden. The evidence developed in this civil proceeding demonstrates that is CRYSTAL clear ...

...

this summary judgement basically says Fox exercised free speech

Defamation is not free speech and doesn't enjoy the protections of the First Amendment which is why you can sue someone for libel or slander.

Fox did nothing wrong in terms of this case

I would look at pages 38-43 of the summary judgement. https://www.documentcloud.org/documents/23736885-dominion-v-fox-summary-judgment