Learn Arabic (Modern Standard): حقيبته في المطار.

Instead of using a separate word like in English, Arabic uses an attached pronoun. We attach the suffix ـهُ (-hu) directly to the end of the noun to mean "his".

The word for bag is حقيبة with a round ة at the end. Why does it look like a regular ت here?

The letter taa marbuuTa (ة) can only sit at the very end of a word. When you add a suffix like the attached pronoun ـهُ (his), the ة has to "open up" into a regular taa (ت) so the letters can connect to each other. That is why it becomes حقيبته.

Where is the word for "is" in this sentence?

Arabic does not use a word for "am", "is", or "are" in the present tense. You simply state the subject, حقيبته (his bag), followed immediately by the location, في المطار (in the airport).

Doesn't في usually mean "in"? Why does the English translation say "at"?

Yes, في literally means "in". Prepositions do not always match up exactly between languages. In Arabic, it is completely natural to say something is "in" the airport (في المطار), whereas English speakers usually say "at" the airport.