Learn Hardcore Russian: Я вежливо поинтересовалась у администратора, работает ли сегодня касса. - I politely asked the receptionist whether the cash desk was open today.

Поинтересовалась means something like inquired / asked politely / took an interest in. It sounds a bit more formal or courteous than спросила, which is the basic verb asked.

So:

спросила = asked
поинтересовалась = inquired, asked in a polite or tactful way

In this sentence, вежливо and поинтересовалась work well together, because both suggest politeness.

Why does поинтересовалась end in -ась?

That ending shows two things at once:

-л- marks the past tense.
-ась shows that the speaker is female.

So:

поинтересовался = a man said this
поинтересовалась = a woman said this

The -сь / -ся part is because the verb is поинтересоваться, which is a reflexive verb.

What does the Я add here? Can it be omitted?