Learn Finnish: Leikkaa nämä oksat!

This is the plural total object. Because this is a specific, completed action (cutting these specific branches entirely), the object takes the total object form. For plural words, the total object looks exactly like the basic nominative plural, ending in -t. If you were saying "cut some branches" (an undefined amount), you would use the partitive plural oksia.

I thought objects in command sentences drop their endings. Why does oksat still have a -t?

You are thinking of singular objects! A singular total object drops its -n ending in a command (like Leikkaa tämä oksa! - Cut this branch!). However, plural total objects always keep their -t ending, even in command sentences.

How do we get leikkaa from the dictionary verb leikata?

Leikata (to cut) is a type 4 verb. To form the command for "you" (singular imperative), you take the "I" form of the present tense—minä leikkaan—and simply drop the -n. Notice the consonant gradation: the single k in leikata gets strong and becomes a double kk in leikkaa.

Why doesn't the word for "these" end in -t like regular plural words do?