Short answer: "Mir ist kalt" (usually no "es") literally means "(It) is cold to me". This is because you feel cold and are not actually cold. To say Ich bin kalt, you are saying either a) you have a cold temperament or b) your core body temperature is cold, in which case you're probably dead and aren't talking anyway. (In an actual example, this means you would say Die Leiche ist kalt with nominative and not Der Leiche ist kalt with dative.)

Long answer:

Wouldn't "mich" fit better as you are the direct recipient of the verb, ist? One thing that a lot of foreign language courses/teachers fail to do is explain exactly what the accusative case does. It expresses a very particular type of object, in most IE languages (like German) dealing with the object of most verbs, motion towards, and duration of time. Even in English, a direct object is very different from a predicative nominative. This is tricky for English speakers, because we usually say "It's me" instead of "It's I" (which sounds archaic because it's an old usage), but this distinction is very present in most other languages. This is a very roundabout way of saying that "to be" doesn't take a direct object, but rather a predicative nominative. You can't be someone. Well, yeah, you can be someone, but not in the same way you can see or find or punch or know someone.

Now the dative, on the other hand, is very versatile. It has varied uses, but they all come down to the idea of reference. In German lessons, the dative usually gets described as the "indirect object" case (which is correct) and then brushed over, so it can get confusing. When I took Latin, it was emphasized much more heavily and described as "to whose advantage or disadvantage something is done or happens", and this definition works pretty well for German, too. If that still doesn't make sense, think of it as "to/for (someone/something)". The dative does serve as the direct object of certain verbs like helfen (this is because such verbs originally had meanings that involved doing something to the object), and as a sort of "secondary" object of other verbs like glauben. With regard to the second category, a lot of verbs can be used with a dative to convey information of who is affected by the information in the sentence. For example, "Mir geht es schlecht" says that it's going poorly for me; "Er erscheint mir als schlau" says that he seems smart to me; Es fällt mir nichts ein says that nothing occurs to me; and so on and so forth.

And why is ist before es? The verb in an independent clause must be in the second position in the sentence, but the other parts can go wherever they want so long as the subject is clear. The es isn't necessary or common in this example, but it's clear that it's the subject, because it's the only nominative noun. You'll see this a lot in various forms, but while we're talking about to be, I should point out that with regard to the "It's me/I" example, Germans usually use this flexible word order to make the phrase "Das bin ich" (lit. "That am I") or "Ich bin's" (lit. "I am it") to me "that's me" or "it's me". Note that das and es are both predicative nominatives in these cases.

Sorry for the long-winded and meandering response. I wasn't sure what was important and what wasn't, so excuse me if this is confusing. I'll clarify any weird bits.