Learn Hardcore Indonesian: Titipan kamu aman; saya mengurusnya sampai kamu kembali. - Your deposited item is safe; I will take care of it until you return.

Indonesian doesn’t use a verb “to be” in present-tense equational sentences. Adjectives can serve as predicates. So Titipan kamu aman literally reads “Your entrusted item safe,” which means “Your entrusted item is safe.”

What exactly does titipan mean and how is it formed?

Base verb: titip = to entrust/leave something with someone (for safekeeping, delivery, etc.).
Noun with -an: titipan = the item/thing that has been entrusted, or the act/arrangement of entrusting.
It can be tangible (bag, key) or intangible (a message: titip salam “send my regards”). Context decides which.

Should it be titipan kamu or titipanmu?

Both are correct.

titipan kamu = separate pronoun; neutral and clear.
titipanmu = clitic -mu; a bit tighter/more informal and common in writing too. Style choice. Many speakers prefer titipanmu for brevity.

What does the -nya in mengurusnya refer to?

It’s the third-person object clitic meaning “it/him/her.” Here it refers back to the entrusted item (titipan). So mengurusnya = “take care of it.” It does not mean “your”; that would be -mu.