Learn Arabic (Modern Standard): هذه هي المرة الخامسة.

Because we are saying "fifth", not "five". الخامسة is an ordinal number. In Arabic, ordinal numbers are actually much easier to use than regular numbers because they follow the exact same rules as normal adjectives.

Since الخامسة acts like an adjective, is that why it matches المرة with ال and a taa marbuta (ة)?

Exactly! The noun المرة (the time) is definite (starts with ال) and feminine (ends with ة). Because ordinal numbers follow adjective rules, الخامسة must match it perfectly, becoming definite and feminine too.

Why is the word هي ("she") in the middle of the sentence?

In Arabic, "is" or "are" is generally implied in the present tense. However, if you just say هذه المرة, it simply means "this time". To say "This is the time", we insert the pronoun هي (literally "she", matching the feminine noun) to act as a separator. It functions just like the "is" in our English translation.

I thought the word for "time" was وقت (waqt). Why do we use مرة (marra) here?

It depends on the kind of "time" you mean! وقت is used for time as a concept on a clock (like "I don't have time"). مرة is used for an instance or occurrence (like "this time", "one time", or "the fifth time").