I know some German but I'm not proficient with it.

It's easier to analyse the sentence by including the subject, typically omitted: "es ist mir kalt" = "it is me cold", or "it's cold to me". It's a lot like saying "that's blue to me", you know? Like, it isn't like you are cold or blue, it's something else, but you're experiencing it. (It's a dative of relation, in both languages.)

“Mir” is German for “me” or “to me” roughly, right?

Roughly, yes. But that gets messy, there's no good equivalent.

Think on it this way: you have a bunch of situations where you'd use the first person, right? English arbitrarily splits those situations between "me" and "I"; German does it between "ich", "mich", and "mir".

That German dative is used in situations like:

• if a verb demands two objects, one gets the dative; e.g. "er gibt mir das Buch" (he gives me the book).
• if the preposition demands it; e.g. "er spricht mit mir" = "he speaks with me"
• if you got a dative of relation (like the above), or benefaction (something done for another person), etc.

I tried to learn some German at some point, but I didn’t manage to learn enough to get comfortable with the various cases.

I got to thank Latin for that - by the time I started studying German, the cases felt intuitive.

But... really, when you're dealing with Indo-European languages, you're going to experience at least some grammatical hell: adpositions (English), cases (Latin), a mix of both (German), but never "neither".

Speaking on Latin, it just clicked me it does something else than the languages you listed - those states/emotions get handled primarily by the verb:

• hungry - esurio (verb, "I'm hungry")
• angry - irascor (verb, "I'm angry")
• cold - frigeo (verb, "I'm chilly/cold")
• scared - timeo (verb, "I fear/have fear")
• brave - fortis (adjective, "strong"); animosus (adjective, roughly "adamant", "stubborn", "angry")