This tells us about students who do not have part-time jobs. But the conclusion is about students who do have part-time jobs. Information about students without part-time jobs can't help us prove anything about students with part-time jobs.

(B) says: home for lunch → part-time job. Combined with the premise (walks to school → home for lunch), we'd get: walks to school → home for lunch → part-time job. That tells us every student who walks to school has a part-time job. But we need to prove the conclusion that some students with part-time jobs do not walk to school. Knowing that all walkers have part-time jobs doesn't tell us anything about whether some part-time job holders are non-walkers.

Same problem as (A). This tells us about students who do not have part-time jobs. We need information about students who do have part-time jobs in order to prove the conclusion.

(D) says some students who don't go home for lunch have part-time jobs. In other words: part-time job ←some→ NOT home for lunch. This bridges the gap. Since some part-time job holders don't go home for lunch, and anyone who doesn't go home for lunch doesn't walk to school (by the contrapositive of the premise), those particular students with part-time jobs don't walk to school. That's exactly what the conclusion says.

Here's the full chain:

part-time job ←some→ NOT home for lunch → NOT walk to school

Therefore: part-time job ←some→ NOT walk to school ✓

(E) says: home for lunch → walks to school. This is actually the reverse of the premise, not something that helps bridge the gap. Combined with the premise, it would tell us that going home for lunch and walking to school always go together. But this still says nothing about part-time jobs, so it can't connect to the conclusion.