Learn Italian: Mio fratello è basso, io sono alto.

You are right that possessives usually take an article! However, there is a special rule for family members: singular family members (like fratello, madre, or padre) do not take the definite article. So it is just mio fratello.

We usually drop words like io in Italian. Why is it included here?

In Italian, subject pronouns like io are usually dropped unless you want to emphasize them or create a contrast. Here, you are directly contrasting yourself with your brother, so including io makes that contrast clear.

How would I use the grammar from this lesson to say that I am taller than my brother?

To make a comparative of majority (meaning more than), you use più and di. You would combine the thoughts into a single sentence: Io sono più alto di mio fratello.

If the speaker is female, does the adjective change?

Yes! Adjectives must match the gender of the person they describe. If a woman is speaking, she would change the ending and say io sono alta. The first part, mio fratello è basso, stays exactly the same because it describes the brother.