Neprestano zaboravljam ključ kod kuće, pa se moram vratiti.

Croatian often leaves out subject pronouns when they are already clear from the verb ending.
Both zaboravljam and moram are 1st person singular, so the sentence already tells you the subject is I. You could add ja for emphasis, but it is not necessary.

Zaboravljam is the imperfective verb, and it fits well with neprestano because the meaning is repeated or habitual: I keep forgetting.
The perfective form zaboravim usually refers to a single completed act, and in the present tense it often has a future-like meaning. So here zaboravljam is the natural choice.

Yes. In Croatian, the present tense is used not only for actions happening right now, but also for habitual or repeated actions.
So Neprestano zaboravljam... means something like I constantly keep forgetting..., even though the form itself is just present tense.

Here ključ is in the accusative singular because it is the direct object of zaboravljam.
But ključ is a masculine inanimate noun, and many masculine inanimate nouns have the same form in the nominative and accusative singular. So it looks unchanged.

Croatian does not have articles like English a and the.
So ključ can mean a key or the key, depending on context. The listener understands which one is meant from the situation.

Kod kuće is the usual expression for at home.
U kući literally means in the house, focusing more on being physically inside a building. In this sentence, the idea is that the key was left at home, so kod kuće is the natural choice.

After the preposition kod, Croatian uses the genitive case.
The noun kuća therefore becomes kuće, so kod kuće is literally something like at the home / by the home, but as a whole it simply means at home.

Neprestano means constantly, continually, or without stopping. It emphasizes that this happens again and again.
Yes, stalno is also very common and would sound natural here. Neprestano can sound a bit stronger or more formal depending on context.

Here pa links the two parts of the sentence and means something like so, and so, or therefore.
It is very common in everyday Croatian for showing a consequence: I forget the key, so I have to go back.

Because vratiti se is a reflexive verb in Croatian.
Many Croatian verbs use se even when English does not use oneself. So vratiti se simply means to return / to go back.

Se is a short unstressed word, called a clitic, and Croatian usually places it near the beginning of the clause.
That is why pa se moram vratiti sounds natural. Even though se belongs logically with vratiti, its position follows Croatian clitic word order.

After moram, the infinitive vratiti se is used because the speaker means one complete act of going back. That is a perfective idea.
If you said moram se vraćati, it would suggest repeated or ongoing returning, like I have to keep going back again and again.