Learn Hardcore Hindi: मुझे दवा से एलर्जी है।

Because Hindi often expresses physical states, feelings, needs, and certain conditions with an oblique/dative-style subject rather than a plain subject.

So instead of saying something structured like I am allergic..., Hindi says something closer to:

To me, there is an allergy from medicine.

That is why मुझे is used, not मैं.

A few similar patterns:

मुझे भूख है = I am hungry
मुझे ठंड लग रही है = I feel cold
मुझे डर लगता है = I feel afraid

So मुझे दवा से एलर्जी है follows a very natural Hindi pattern.

What exactly does मुझे mean here?

मुझे is the oblique form of मैं used with certain postpositions and constructions. In this sentence, it means something like to me.

So:

मैं = I
मुझे = to me / me

In many everyday sentences, मुझे is used where English would simply use I:

मुझे पता है = I know
मुझे खुशी है = I am happy
मुझे दवा से एलर्जी है = I am allergic to medicine

Why is से used after दवा? Doesn’t से usually mean from or ?