Learn Polish: Zawsze chowam swój prywatny dziennik do swojej szafki. - I always hide my private diary in my own cabinet.

The pronoun swój means "one's own" and refers back to the subject of the sentence. Since the subject is "I" (ja), you are hiding your own diary in your own cabinet. While using mój would be understood, Polish strongly prefers swój whenever the object belongs to the subject performing the action.

Since dziennik is the direct object of the verb, why doesn't swój change to swojego?

The verb chować takes the accusative case for its direct object. However, dziennik (diary) is a masculine inanimate noun. For masculine inanimate nouns, the accusative case looks exactly identical to the nominative (dictionary) form. If you were hiding a masculine animate noun, like a dog (pies), it would change to swojego psa.

Why does the second pronoun take the form swojej?

The preposition do (into/to) always requires the genitive case. Because szafka (cabinet) is a feminine noun, the pronoun must take the feminine genitive form, which is swojej.

Why is the imperfective verb chowam used rather than the perfective schowam?

The adverb zawsze (always) is the key here. It tells us that this is a repeated, habitual routine. In Polish, you must use imperfective verbs for habits and regular actions, whereas perfective verbs are reserved for completed, one-time events.