Learn Finnish: Kirjoitan päiväkirjaani usein.

The 1st person ending -n on the verb kirjoitan already tells us who is doing the action, so the pronoun minä (I) can be dropped. This is standard practice in written Finnish.

Why isn't the word minun (my) used before the diary?

The possessive suffix -ni at the end of päiväkirjaani already means "my". For 1st and 2nd person possessives (my, your, our), the suffix alone is enough, and you can simply drop the pronoun minun.

Why does Finnish use the illative (into) case for "in my diary"?

English says you write "in" a diary, but Finnish expresses this as putting words "into" a book. Therefore, it uses the illative case, which shows movement into something.

How exactly is the word päiväkirjaani formed?

First, you add the illative ending -an to päiväkirja to get päiväkirjaan (into a diary). However, when you add a possessive suffix like -ni, the final -n of the case ending is dropped. So päiväkirjaan becomes päiväkirjaa- + -ni = päiväkirjaani.