Learn Hardcore Korean: geonganghan momeul wihae maeil undonghaeyo.

Korean often drops the subject when it’s obvious from context. Here, it’s understood that you’re talking about yourself (“I”). If you want to be explicit, you can say:
저는 건강한 몸을 위해 매일 운동해요
but even without 저는, the meaning stays clear.

What does 건강한 mean and how is it formed?

건강한 is the attributive (“modifying”) form of the descriptive verb 건강하다 (“to be healthy”). To turn a descriptive verb into an adjective that modifies a noun, you:

Drop -다 from 건강하다 → 건강하-
Add -ㄴ (because the verb stem ends in a vowel) → 건강한
So 건강한 몸 literally means healthy body.

What does 을 위해 mean and how does it function in this sentence?

The pattern X을/를 위해 means “for X” or “in order to achieve X.” It marks the purpose or goal.
• 건강한 몸을 위해 = “for a healthy body”
By attaching 을/를 to 건강한 몸 and following it with 위해, you indicate purpose: “(I exercise) in order to have a healthy body.”

What’s the difference between 을 위해 and 위해서?

Both 을 위해 and 을 위해서 carry the same basic meaning (“for the sake of…”). The difference is mostly stylistic:

을 위해 is slightly more concise and common in written or formal contexts.
을 위해서 is a bit more colloquial and often used in speech.
You can freely substitute them:
• 건강한 몸을 위해 매일 운동해요
• 건강한 몸을 위해서 매일 운동해요
Both are correct.

Why is it 운동해요 instead of 운동을 해요? Can you use both?

운동하다 is a compound verb (Sino-Korean noun 운동 + 하다). When you conjugate it, you typically write it as one word: 운동해요. However, you can also treat 운동 as a separate noun and say 운동을 해요, adding the object marker 을. Both forms are acceptable and mean the same thing: “do exercise.”

What is 매일, and what’s the difference between 매일 and 매일마다?

매일 means “every day.” It’s the most common time adverb for a daily routine.
매일마다 also means “every single day,” but it’s more emphatic or poetic and less frequent in everyday speech.
You’ll almost always hear native speakers say 매일 운동해요 rather than 매일마다 운동해요.

Why is the verb ending 해요, and could we use 합니다 instead?

해요 is the present-tense, polite (요-ending) form of 하다. It’s used in everyday polite conversation.
If you want a more formal or written tone, you can use the 하십시오체 form 합니다:
• 건강한 몸을 위해 매일 운동합니다
That’s more appropriate in announcements, presentations, or very polite contexts.

Can I change the word order? For example, 매일 건강한 몸을 위해 운동해요 or 매일 운동해요, 건강한 몸을 위해?

Korean word order is relatively flexible, but the verb still goes at the end. Time adverbs (like 매일) usually come early, and purpose phrases (like 건강한 몸을 위해) can come before or after the time expression:
• 건강한 몸을 위해 매일 운동해요
• 매일 건강한 몸을 위해 운동해요
Both are grammatically fine. Splitting them into two clauses (매일 운동해요, 건강한 몸을 위해) sounds choppy for a single idea; learners usually pack them together as one sentence.

Could I shorten 건강한 몸을 위해 to 건강을 위해?

Yes. Native speakers often say 건강을 위해 매일 운동해요, literally “I exercise every day for (my) health.” It’s more concise and idiomatic, with 건강을 위해 (“for health”) standing in for “for a healthy body.”