Learn Danish: Udgiveren bliver ved med at lede efter en god oversætter. - The publisher keeps looking for a good translator.

They are both formed by adding the -er suffix to a verb to describe a person who does that action. Udgiver comes from the verb udgive (to publish), and oversætter comes from oversætte (to translate). Since these verbs already end in -e, we just add an -r.

Why does udgiveren end in -en?

The -en ending makes the noun definite, changing it from 'a publisher' to 'the publisher'. All agent nouns ending in -er that describe people are common gender (en-words), so they always take -en in the singular definite form.

How does the phrase bliver ved med at work here?

This is a very common fixed phrase meaning 'keeps doing' or 'continues to'. The base expression is at blive ved med (literally 'to stay at with'). You follow it with at and the infinitive of the next verb, which in this sentence is lede.

Could I use søger instead of leder efter for 'looking for'?

You could, but at lede efter is the most natural, everyday way to say you are actively looking around for a physical thing or a specific person. At søge is usually reserved for abstract things, like applying for a job (søge et job) or searching a database.