Learn Dutch: Waarom heb jij je bemoeid met hun gesprek? - Why did you interfere with their conversation?

The main verb here is zich bemoeien (to interfere/meddle), which is a reflexive verb. In Dutch, all reflexive verbs use hebben as their helper verb in the present perfect tense. You will never use zijn with a reflexive verb.

Why are jij and je right next to each other?

Jij is the subject ("you"), and je is the reflexive pronoun ("yourself"). Because this is a question starting with a question word (Waarom), the verb (heb) comes before the subject. The reflexive pronoun stays right after the subject, putting them side-by-side: heb jij je... (literally "have you yourself...").

Why does the past participle bemoeid end in a -d and not a -t?

This follows the 't kofschip rule. The infinitive is bemoeien. If you remove the -en, you get the stem bemoei, which ends in a vowel (i). Since vowels are not consonants in the word 't kofschip, the past participle gets a -d.

Does the verb bemoeien always use the preposition met?