Learn Dutch: Ik ben bezig met de schoenen die versleten zijn. - I am busy with the shoes that are worn out.

This is a restrictive relative clause (beperkende relatieve bijzin). It tells us exactly which shoes you are busy with (the worn-out ones, not the new ones). Because this information is essential to identifying the shoes, we do not separate it with a comma.

How would the meaning change if I put a comma before die?

If you wrote Ik ben bezig met de schoenen, die versleten zijn, it would become a non-restrictive clause (uitbreidende relatieve bijzin). This means you are busy with the shoes in general, and you are just adding a casual side note that they happen to be worn out.

Why do we use die here instead of dat?

The relative pronoun refers back to de schoenen. Because schoenen is plural, we must use die. You only use dat to refer back to singular het-words.

What is the infinitive of versleten?

The infinitive is verslijten (to wear out). Here, versleten is the past participle, which functions like an adjective to describe the current state of the shoes.

Is it always bezig met when you are working on something?