It's true, if you are talking about money in general, you don't need a "the", but that's true of any object. You use "the" to signify a specific object, but not when referring to it in general. Consider:
"Do you carry any keys?"
"Will you grab the keys off the coffee table?"
That's probably not surprising to you that when talking about keys in general, you don't use "the", but when talking about specific keys, you do. Keys are really no different from money. Consider:
"Do you carry any money?"
"Will you grab the money off the coffee table?"
The second sentence might be abbreviated if it's expected we already know where the keys or money are:
"Will you grab the keys?"
"Will you grab the money?"
The second sentence might sound strange, but that's just because we normally don't care that you have some particular money, but just some money in general. Imagine you were leaving to make a deposit and had an envelope of cash on the table for this specific purpose. In that context, the sentence seems fine, and would even be strange without a "the".
This abbreviation is what is happening in your example. She's saying she needs the money from the job, which is why she needs the job. But wait, does she really need that money in particular? No, of course not, she certainly just needs money in general. So why specify? A response in either form would be grammatical, but there is a slight difference in implication. Saying "I need money" doesn't relate back to the job you're applying for. It's a reason to apply for a job, sure, but why this particular one? All jobs pay money. In contrast "I need the money" is emphasizing that you want the money this particular job will supply. It's the difference between saying "I need money and this job happens to pay money, so I applied for it" vs "I need the money this job will pay." If you leave out the "the" then the sentence abbreviates to the first example, which is not a very good response in an interview, regardless of how true it probably is.
Keep in mind that although this example has a "the" because the source is specified, it can be specified by other means. The important thing is that you are not referring to money in general, but some specific money.