“Se le descargó la batería” looks like an extension of the idea of the sympathetic dative, which is normally found with people and their inalienable possessions. I have no idea whether a linguist would say it's a proper sympathethic dative, but it follows the principle.
The verb is pronominal intransitive descargarse "to discharge; to become discharged". The pronoun le is the third person singular indirect object pronoun, and refers to the car.
As I said this looks like a sympathetic dative but it might be only similar to one, because it's only borderline compulsory to use it. It's not un-idiomatic to use the possessive pronoun instead in this case, so the following alternative sounds OK:
El carro no anda porque se descargó su batería.
Contrast with a certified sympathetic dative with a similar structure, like “Él usa peluca porque se le cayó el pelo”; nobody would naturally say
❌ Él usa peluca porque se cayó su pelo.
So the structure is like a sympathetic dative, but possibly because we're talking about an inanimate object and a mere mechanical part that can be detached, the connection is not felt that strongly and we can also use the basic, non-sympathetic syntax instead. (I still find the sympathetic form more natural in this case, and that is how I would express the idea.)