Both are used, but the narrative of this video is written in the present tense, as if the action is happening in the moment. Consider the next sentence:
The little owl is sleepy and gets ready for bed.
This use of the present tense in narrative is not uncommon in children's literature, possibly to teach young readers about things that are common, repetitive, and habitual, or possibly just so they can enjoy it as if they are part of (or at least viewing) the action as it happens.
I found this quote useful:
The present tense is often used in simplified narrative. It has to be simplified, because there is only one tense and the sequence of actions can only be represented by placing them one after another as they happen. source
By convention there's a very different feeling between using the simple present and the present progressive to describe action. The simple present tense tells events as they happen, one after the other, and can be somewhat calm and relaxing. It's often used in things like jokes, where you're waiting to hear what happens next.
A piece of string goes into a bar. The bartender yells, "Hey, we don't serve string here!"
So the string goes back outside, ties himself up, and pulls his ends apart. He then goes back in.
"Didn't I tell you we don't serve string here?!" says the bartender.
"No, I'm a frayed knot," says the string.
The progressive tense, on the other hand, describes what is going on, leaving each event unresolved. It is an uncomfortable way to write narrative, and only rarely used for poetic effect:
See the lights of a neighbor's house
Now she's starting to rise
Take a minute to concentrate
And she opens up her eyes
Moving into the universe and she's
Drifting this way and that
Not touching the ground at all and she's
Up above the yard
"And She Was" by the Talking Heads
Even this song uses the past progressive most of the time, alternating with some simple present.