Breakdown of Der Ärmel ist schmutzig, weil ich dir die Schnürsenkel zubinde.
ich
I
sein
to be
dir
you / to you [informal singular, dative]
weil
because (subordinating)
Questions & Answers about Der Ärmel ist schmutzig, weil ich dir die Schnürsenkel zubinde.
Why does this sentence use dir die Schnürsenkel instead of deine Schnürsenkel?
This is the possessive dative (Dativus possessivus). When talking about things closely associated with a person, like body parts or clothing (in this case, shoelaces), German prefers using a dative pronoun (here, dir) to show who is affected, plus the definite article (die) instead of a possessive determiner (deine). While deine Schnürsenkel isn't grammatically wrong, dir die Schnürsenkel sounds much more natural.
The English translation says 'The sleeve is dirty' but could also imply 'My sleeve'. Why doesn't the German use Mein Ärmel?
Similar to the possessive dative rule for someone else's clothing, German often drops the possessive pronoun (mein) for your own clothing or body parts when the context makes ownership completely obvious. Since the speaker is the one doing the tying and getting dirty, Der Ärmel naturally implies the speaker's own sleeve.
The verb is zubinden (to tie). Why is it written together as zubinde at the end of the sentence instead of being separated?
zubinden is a separable verb (zu + binden). In a normal main clause, the prefix zu would separate and go to the end. However, because weil introduces a subordinate clause, the conjugated verb is pushed to the very end of the sentence. When a separable verb is at the end of a subordinate clause, its prefix and verb rejoin into a single word: zubinde.