Learn Russian: Ты должен беречь себя.

себя is a reflexive pronoun that always refers back to the subject of the sentence, no matter who that subject is. Since the subject (Ты) is doing the action to themselves, we use себя (meaning "yourself" here). It is in the Accusative case because it is the direct object of the verb беречь.

Why is the word for "must" (должен) acting like an adjective?

In Russian, "must" or "have to" is expressed with должен, which is a short adjective rather than a verb. Because it is an adjective, it changes to match the gender of the subject. Here, the speaker is addressing a male, so they use the masculine form должен. If they were speaking to a female, it would be должна.

Why doesn't the verb беречь change to match Ты?

Whenever you use должен to say someone "must" do something, the action verb that follows it stays in its infinitive (dictionary) form. It works just like English: you say "must take care", not "must takes care".

I noticed беречь ends in -чь. Don't most Russian infinitives end in -ть?

You are exactly right! Most Russian verbs end in -ть (like делать or говорить). However, a small group of verbs ends in -чь (like мочь). You do not have to worry about conjugating it here because it stays in the infinitive, but be aware that verbs ending in -чь usually have some spelling changes in their root when you do conjugate them.