"Mir wird kalt." - I don't understand the grammar.

I don't study German, but the little I do know from music (Rammstein and Eisbrecher being my main teachers) this is a normal thing.
For example, in Eisbrecher's self-titled song (on their self-titled album no less ) the vocalist repeatedly says "...und es wird kalt"
Similarly, in a Rammstein song, they sing: "Mir ist kalt."
I think it's just one of those random things of the language. I suppose the phrase could be written "es wird mir kalt" (sorry if that's a weird word order) or something like that, so it would translate as "It's becoming cold to me." (I believe the dative case equals an indirect object)
Maybe it shows that the coldness is related to the speaker, not necessarily the actual surroundings?
I know that's a ramble, but I hope it helped in some small way

In German you say "mir ist kalt" for "I am cold", so "mir wird kalt" just means "I'm getting cold".

"Ich bin kalt" would rather mean something like "I'm cool/untouched".

Hope this helps..

Yeah, Clochette is right. And "es wird mir kalt" is a good explanation, too.
The other case, I imagine "Ich bin kalt" would be right, is when one is freezing to death (in an avalanche or alive in a coffin?)
by the way: "jemanden kalt machen" - "kill someone"

dClochette and Mauler are correct. I cant remember the grammar enough to give a great explanation but when you say mir is kalt you are really saying es wird mir kalt. Since Es is taking the subject form in the sentence you use the dative pronoun mir. Trying to teach yourself German from the music is great and will really help your vocab, but German grammar is crazy complicated (der die das des den dem usw. )

mir = to me or of me , mir wird kalt is I am getting cold, as has been said before.

One thing I'd k´like to add. Mir ist kalt is special to the german language because it doens't have a subject. The german subject is always a nominativ, you can find it by asking who. So when you ask who is cold you would need to answer (the correct grammar) I (Ich, first person singular). But the sentence I am cold (Ich bin kalt) has a different meaning because it doesn't convey that you feel the cold for yourself, it just can be a logical consequence. So we take a reflexive pronoun to change the information, though we don't use the pronoun that it referres to). You can use the other german one mich as well, you just have to change the case from dativ to akkusativ (Es friert mich --> mich friert es --> mich frierts) The "Es" looses its meaning in this sentence and becomes a substitute for the missing subject. It's a great example to show that the language develops and the grammar trys to describe it and not the other way round ^^. I hope you didn't got too bored, if you have questions feel free to ask, if I did mistakes, feel free to correct me, I still have to learn a lot about the english language.

I think "wird" is in third-person, because it's somehow like in "it 's raining" (what/who is raining?- It is...)
Mir wird kalt is a short version of Mir wird es kalt

I wonder if the same thing applies in a popular biblical passage: "Es werde licht." (Let there be light).

No, it means literally "It becomes light". In this case light is an adjective.

Thanks for explaining.