Learn Portuguese (Portugal): Ela comeu meia tarte de maçã.

Because here, meia acts as an adjective meaning 'half of a', modifying the feminine noun tarte. Adjectives must agree in gender and number with the noun they describe. If it meant 'kind of' or 'a bit' (like being 'kind of' tired), it would be the invariable adverb meio. Since she ate a physical half of a pie, it changes to match the feminine noun.

Notice how English says 'half an apple pie'. Why isn't there a word for 'an' in the Portuguese sentence?

In Portuguese, when you use meio or meia as an adjective to mean 'half a', you place it directly before the noun. You do not add the indefinite article (um or uma) in between, so it is simply meia tarte.

Why is it de maçã and not da maçã?

When describing a type, flavor, or material of a food item, Portuguese usually uses just the preposition de without a definite article. It is an apple pie in general, not a pie made from a specific, previously mentioned apple.

Could I use metade to express this same idea?

Yes, you could say Ela comeu a metade de uma tarte de maçã. However, using meia directly before the noun is much more concise and common in everyday spoken Portuguese.