Learn Polish: Kobieta wzięła swój parasol, płaszcz i telefon. - The woman took her umbrella, coat and phone.

Swój means "one's own". Because the subject of the sentence (the woman) is the owner of the items, you use swój to say she took her own things. If you used jej, it would imply she took another woman's umbrella!

Why is the form exactly swój here, and not swoją or swoje?

The pronoun must match the noun it describes in gender and case. The direct object here is parasol (umbrella), which is a masculine inanimate noun. In the Accusative case (used for direct objects), masculine inanimate nouns and their adjectives look exactly the same as they do in the basic Nominative dictionary form. Therefore, it stays swój.

Does swój apply to the coat and phone as well, or just the umbrella?

It applies to all three! In Polish, if you have a list of nouns that share the same gender and case—like parasol, płaszcz, and telefon (all masculine inanimate in the Accusative case)—you only need to use the pronoun once at the beginning of the list.

I see wzięła means "took". Why not use brała?

Wzięła is the completed (perfective) form of the verb. You use it here because she completed the action of picking up her things. Brała is the ongoing (imperfective) form, which would mean she was in the middle of taking them or took them repeatedly.