Milyen szabály szerint kell betenni a "that"-et?

When you don't use a comma (in other words when it is a defining relative clause), you can use "that".

When you put a comma (in other words in a non-defining relative clause), you must use the proper pronoun (or you can omit it and use the participle).

Your sentence falls into the first category and as such you could say it with "that" or "who" as well. That is considered a bit slangisher by many.

An example where you can't use that would be:

"I met my father yesterday, who just came back from Italy." Of course, this is assuming you only have one father. Had you two, then this could be a defing relative clause, in which case there would be no comma and "that" could be used as well.