Learn French: J'ai expliqué que mon beau-frère cherchait du travail. - I explained that my brother-in-law was looking for work.

This is the sequence of tenses in action! The original statement was in the present tense: "Il cherche du travail" (He is looking for work). Because you are reporting this using a past tense verb (J'ai expliqué), French grammar rules require you to shift that original present tense back to the imparfait (cherchait).

Could I use the passé composé (a cherché) for the second verb instead of the imparfait?

Only if you want to change the meaning. If you used the passé composé (a cherché), it would mean the job hunt was already over when you explained it: "I explained that he had searched for work." To show that he was looking at the time you were explaining it, you must use the imparfait.

Can I drop the word que just like we drop "that" in English?

No. In English, we often drop "that" and simply say "I explained my brother-in-law was looking for work." In French, the conjunction que is strictly mandatory to connect the two parts of the sentence and can never be omitted.

Why is it du travail rather than un travail?