There's one of his maxims I like better: You don't know something until you can prove it 3 different ways.
In mathematics, it begs the question of what "understanding" really means. To understand an object doesn't necessarily mean divining an unquestionable structure by chance. What usually happens is that you're investigating some kind of problem (usually with real-world applications, if distant), and then you find you need a certain tool or a certain theory to simplify your problem, make it more tractable or more abstract. For example, you could be studying permutations, and from there the Binomial comes naturally, as well as the factorial function. From this theory, comes several definitions. The definitions are such to further you goal: they are the ones that make your tool easier to use, simpler, more "streamlined", more suitable to approach your application with minimal special cases. This is how something like '0!' is defined, and how most theories are discovered.
The thing about understanding is that it's a bit too much to require to "understand" something (even to yourself).
How can one know when he's reached "understanding"? Being used to it should be good enough for most purposes. If you know the rules, and you know how to apply them, that's mathematics.
Perhaps another direction to understanding is seeing a thing from a variety of lenses (connecting to different fields), expanding your ability to apply a tool, seeing more broadly. That's when you generalize, and you're able to see what your had as a special case (another definition of understanding): from addition to algebra, to rings, to abstract algebra. From numbers to equations to functions, each step perhaps you "understand" the fundamentals better by having a broader perspective on generalization. But of course that's only useful if your generalization is useful at all.