Learn Hardcore Russian: Я увлекаюсь спортом и бегаю каждое утро. - I am keen on sports and run every morning.

The verb увлекаться is a reflexive (or middle‐voice) verb meaning “to be passionate about,” “to take a keen interest in.” The suffix -ся doesn’t make it passive here; it turns увлекать (“to enthrall, carry away”) into “to get carried away by something” (i.e. “to be really into it”).

Why is спортом in the instrumental case?

In Russian, verbs like увлекаться, заниматься, интересоваться require the instrumental case for the thing you’re into or occupied with. So спорт becomes спортом to show “I am passionate about sports.”

How is увлекаться спортом different from заниматься спортом or любить спорт?

заниматься спортом = “to practice/do sports” (focus on the activity itself)
любить спорт = “to love sports” (focus on the feeling)
увлекаться спортом = implies a hobby or deep passion; you regularly get carried away by sports, it’s more than just liking or doing it.

Why is бегаю used here? Can we use побегаю instead?

бегаю is the 1st‐person‐singular present tense of the imperfective verb бегать (“to run”). It expresses a habitual or repeated action: “I run (every morning).”
The perfective побегать has no present tense; я побегаю means “I will run (for a while).” You cannot use it to describe a current habit.

Could you explain каждое утро? Is that accusative or nominative, and can I say по утрам instead?

каждое утро literally “every morning.” As a time‐expression it’s in the accusative (though neuter nominative/accusative look identical).
You can also say по утрам (“in the mornings”), which is по
- dative plural and equally expresses a habitual “every morning.”

Why is the subject Я included? Can it be omitted?

Russian is a pro‐drop language, so the pronoun я can be omitted because the verb endings already show who’s doing the action.
• With я: Я увлекаюсь спортом и бегаю каждое утро.
• Without я: Увлекаюсь спортом и бегаю каждое утро. both are correct; including я adds emphasis or clarity.