Which one is correct? "Explain me" or "Explain to me"?

"Explain me" in the context you probably are thinking of, would be pidgin English.

The only way it would make sense if it meant that "me" was the thing you want explained. "Explain mathematics", "Explain cars", "Explain me".

"Explain to me" is perfectly fine, either as part of a sentence...

Explain to me why you did that.

... or with the rest of the sentence provided by context.

I don't understand why you did that. Explain to me.

Dear Dr. Freud,

I am having an identity crisis. I no longer understand myself.
I would like to book an appointment with you so that you can explain me to myself.

yours sincerely,
Mr. Smith

Or

Child: do you like my picture?
Parent: It's lovely! Why don't you explain to me what it is?

It appears to depend on the context.

Explain is a transitive verb (it take an object), so you can say "please explain cartography", or "explain apples", etc. It is also a ditransitive* verb, so it can take two objects,

e.g.

explain to the tree what it is

object 1: the tree
object 2: it

explain to him how to do it

object 1: him
object 2: it

Which is just like how you might use show

show [to] me what it is

vs.

show what it is to me

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*Thanks to FumbleFingers for the info

The meaning of "correct" here is a bit slippery. People usually say explain to me...

I don't think the word "to" is either grammatically or logically necessary - we don't normally bother with it in give me, show me, lend me for example. But given we normally do include the preposition in explain to me, OP should be aware that some people will think he's ill-educated if he doesn't follow the majority usage.

EDIT: Being in "chart mode" here, I'll just include a related construction give it [to] me, where this chart clearly shows how usage has shifted over the past couple of centuries to favour including the preposition...

As with explain, the primary argument for including "to" is simply that this is what most other people do. And the primary argument against doing it is that other people tend to think if you don't copy the most common usage, you're not a competent speaker.

Depending on the context, explain me is as correct as tell me. Both make perfect sense to me.

Slim's answer is good. Let's build on it a bit – mind the punctuation:

Explain it to me.
Correct.

Explain me.
Incorrect / You won't hear it.

Explain me this. Correct.

Can you explain me how to get there?
Correct.

Now, if we want to get grammatical, we can dissect the sentence to analyze what's going on in terms of structures. But, really, that is optional. The short story is that most of them are correct because you'll hear them and they mean something, while the incorrect one is never heard and doesn't really mean anything.

If we want to analyze it… Then the key is that it's a transitive verb, i.e.: it needs an object, which will be the thing being explained. Let's generalize:

Explain something to someone.

But you won't say or hear:

Explain someone.

(Well, you could say it, but that would be the exception rather than the rule. e.g.: Explain Chopin to me. if you talk to a pianist who knows that composer really well.)