Learn Hardcore Korean: olhae mokpyoneun maeil hangugeo daneoreul oe-uneun geoyeyo. - This year’s goal is to memorize a Korean word every day.

In Korean, -는/은 marks the topic—what you’re talking about—while -이/가 marks the subject—what performs the action or exists. Here, you’re introducing 올해 목표 (“this year’s goal”) as the topic of the sentence. You’re not saying “the goal does something,” but “as for this year’s goal, it is …”

What is the function of -는 in 외우는 거예요? Why not just 외워요?

The -는 ending is the present attributive form (relative clause marker). It turns the verb 외우다 (“to memorize”) into 외우는 so it can modify 거예요 (“thing/is”).
Structure:

외우다 (verb)
외우 + 는 = 외우는 (attributive)
외우는 거예요 = “It’s the thing of memorizing…”

Using 외워요 would simply be “(I) memorize,” not “the act of memorizing is my goal.”

What does 거예요 mean here, and why not use 것이에요?

거 is a contraction of 것 (“thing”). In polite speech, 것이에요 naturally shortens to 거예요. Both mean “it is a thing,” but Koreans almost always say 거예요 in conversation.

Could I say 올해 목표는 매일 한국어 단어 외우는 거예요 without 를 after 단어?

No. 를 is the object particle required because 단어 is what you’re memorizing. Omitting 를 would make the sentence ungrammatical or ambiguous. Always attach 를/을 to mark the direct object of a verb.

Why is 매일 placed before 한국어 단어를 외우는 거예요? Can I move it elsewhere?

매일 is a time adverbial (“every day”) and Korean is flexible, but the most natural order is:
(Topic) + Time + Object + Verb phrase.
So 올해 목표는 (topic), 매일 (time), 한국어 단어를 (object), 외우는 거예요 (verb phrase).
You could say 올해 목표는 한국어 단어를 매일 외우는 거예요, but putting 매일 right after the topic is more common.

What’s the difference between 외우는 거예요 and 외울 거예요?

외우는 거예요 uses the present attributive -는
- 거예요, focusing on the act of memorizing itself (“It is memorizing …”).
외울 거예요 uses the future marker -을
- 거예요, meaning “I will memorize …” as a future plan.
  In this sentence, you’re defining your goal (a noun-like idea), so 외우는 거예요 fits better than the simple future tense.

Why not use 하기 as in 외우기 instead of 외우는 거예요?

You could say 올해 목표는 매일 한국어 단어를 외우기예요, but that sounds awkward because -기 is a nominalizer and you’d normally follow it with 로 (외우기로). A more natural alternative is:
올해 목표는 매일 한국어 단어를 외우기로 했어요 (“I’ve decided to memorize Korean words every day”).
However, to simply state your goal, 외우는 거예요 is the most idiomatic.