Learn Hardcore Hindi: ओह, मुझे सिरदर्द है।

Because Hindi usually expresses things like pain, hunger, thirst, and many physical or emotional states with an experiencer construction rather than I + have.

So instead of saying something literally like I have a headache, Hindi says something closer to:

मुझे सिरदर्द है = To me, there is a headache / I have a headache

Here:

मैं = I as the subject
मुझे = to me / me

This pattern is very common in Hindi:

मुझे भूख है = I am hungry
मुझे प्यास है = I am thirsty
मुझे ठंड लग रही है = I feel cold

So मुझे is exactly what you expect in this kind of sentence.

What exactly does मुझे mean here?

मुझे is the oblique/object form of मैं and here it means to me or me.

A helpful way to think of it:

मैं = I
मुझे = me / to me

In this sentence, मुझे marks the person experiencing the headache.

You will see मुझे in many other sentences too:

मुझे हिंदी आती है = I know Hindi
मुझे वह पसंद है = I like that
= I am feeling scared