You've already received a lot of good responses, but let me see if I can fill in some gaps that I didn't see in the other replies.

In a situation like this, it's okay to let the other person know you're feeling nervous and that's making it harder to recall things. I'm sure that professor knows what it's like to be in that situation, but unless you remind him in the moment he may not realize you feel that way right then. Some people will even try to be accommodating. And if the person decides to be a dick about it, then you know that you probably don't want to work with them. So it's typically a win-win to mention it when you're in these one on one sort of situations.

So much of mathematics is about knowing the definitions and the major results. I had the good fortune early in undergrad of having a math professor tell us how to do our homework. He said, for each problem you sit down to do write out each definition that the problem statement uses. Try to write it from memory, but it's okay to look it up if you get stuck and you should double check that you recall it correctly. Doing this will build your ability to recall the definitions, even if it seems boring. Actually, boredom in this sense is a good signal that you're starting to master recalling it.

The next thing he said to do, as it becomes less effort to recall the definition, is to start thinking about what the definition means. What ideas does it capture? What is something that fits the definition and what is something that doesn't fit the definition?

He also said, often times with undergraduate text book problems, they can be solved quite readily when the definitions are sitting there on the page next to each other.

In many ways, understanding a subject of math can be reduced to the combination of knowing the definitions and their implications (the theorems/lemmas about them). Once you've studied mathematical reasoning for a while, many proofs are sort of rote in the sense that they follow from the definitions. However, some theorems tend to embody some sort of insight or cleverness. Those tend to be named theorems. For those, it's fine to just memorize the cleverness. You should at least know their implications and the major insight behind their proof. The rest of the details are hopefully the small rote reasoning steps that you can reconstruct as needed.

Another tip, is to take the things you felt you blanked on or wished you had known and study up on those before you talk to him again. It's a "better late than never" situation. And you could then tell him you reviewed it and you'd like a second chance at answer some of his questions. See what he says. Ideally, he'll see it a sign of maturity and willingness to improve yourself.