Learn Hardcore Korean: jeoneun yeolsoereul jaju ilheobeoryeoyo.

저 = I (humble) + 는 = topic marker. It marks “I” as the topic of the sentence (as for me), indicating that the action (losing keys) is about you. In spoken Korean, you can omit 저는 if the context already makes it clear who you are talking about.

Why is 열쇠 followed by -를 (열쇠를)?

-를 is the object marker (use -을 after a noun ending in a consonant). Korean follows a Subject-Object-Verb word order, so the thing being acted on (your keys) must be marked. 열쇠를 shows that the keys are what you lose.

Where does the adverb 자주 go in this sentence, and why?

Adverbs modifying verbs or adjectives typically appear right before the verb or adjective. Here, 자주 modifies 잃어버려요, so you place it immediately before the verb:
저는 열쇠를 자주 잃어버려요.
(You could also say 저는 자주 열쇠를 잃어버려요 for slight emphasis on frequency, but it’s less common.)

What’s the difference between 잃어요 and 잃어버려요?

Both derive from the verb 잃다 (to lose), but:

잃어요 is the simple present polite form of 잃다 – just to lose.
잃어버려요 comes from the compound 잃어버리다, which adds a sense of complete, often accidental or regrettable loss (you can’t easily get it back). For items like keys, you almost always use 잃어버려요.