“She was asking something about her travel arrangements, but I didn't quite get what she wanted.”
You got a confusing e-mail from someone at work. She asked a question, but you didn't understand what she wanted to know. You say this while you're explaining the situation to your boss.
She was asking something about her travel arrangements, but I didn't quite get what she wanted.
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In the situation above, the speaker could use a simpler phrase:
She was asking about her travel arrangements.
But this sentence doesn't express the feeling of confusion. If you want to explain what someone asked and show that you're confused, use "ask something about ___".
"Travel arrangements" are the plans that you make for a trip. They include booking a flight, renting a car, and reserving a hotel room. The phrase "travel arrangements" is usually used in the context of a business trip rather than a personal vacation.
To "get" something means to understand it. For example, when you didn't understand something, but then your friend explains it to you in a way that you do understand, you can say:
Oh, OK. I get it now!
The phrase "I didn't quite get ___" means that you almost understood something, but you didn't actually understand it. The words "not quite" mean "almost, but not really". Another example of "not quite" is:
I'm not quite sure what I want to do after college.