Learn Turkish: Onun kardeşi yeni bir işe başvuruyormuş. - I heard his sibling is applying for a new job.

It combines two suffixes on the verb başvurmak (to apply). The -uyor part is the present continuous tense (is applying), and the -muş part is the inferential suffix used for hearsay. Together, they mean that an ongoing action is happening, but you are reporting what you heard rather than what you saw directly.

Why is the dative case used in işe instead of just iş?

In English we say 'apply for a job', but in Turkish the verb başvurmak requires the dative case (-e or -a suffix). You are structurally saying that the sibling is applying 'to' a new job, so iş becomes işe.

Could I say bir yeni işe instead of yeni bir işe?

Generally, no. In Turkish, when a noun has both a descriptive adjective like yeni (new) and the indefinite article bir (a/an), the adjective comes first. The standard word order is almost always Adjective + bir + Noun.

Since kardeşi already means 'his sibling', why do we also need onun at the beginning?

You are right that kardeşi alone means 'his sibling' or 'her sibling' because of the -i possessive suffix. Adding the pronoun onun is optional. It is used here just to add emphasis or to make it absolutely clear who you are talking about in the broader conversation.