I want to preface this by saying I'm a graduate student in pure math and I can only speak for pure math. I don't know what level of understanding is appropriate for applied math, and I'm sure that can vary depending on what kind of applied math.
I would say that to have a solid understanding of things at the basic graduate level, knowing just the concepts or the general idea is not enough. It is best to be able to prove most of the math that you use.
You don't need to memorize every step of a proof, that's too much and it's not really useful anyway. It's better to have the ability to quickly recollect the proof on your own. To do this, it's good when you're reading a proof to keep on the lookout for things that you wouldn't have come up with on your own, and do memorize those parts.
Besides, those difficult techniques or counterintuitive perspectives will often show up in other proofs too, or sometimes in whole fields of mathematics.
Example: in real analysis, do you remember how to prove that if $x_n, y_n$ are sequences which converge to $x, y$, then $x_ny_n$ converges to $xy$? You do so by splitting up the difference: $$|x_ny_n - xy| = |(x_ny_n - xy_n) + (xy_n - xy)| \leq |x_n - x| \cdot |y_n| + |y_n - y| \cdot |x|$$ And now you use the fact that since $y_n$ is a convergent sequence, it is bounded, and hence $|x_n - x| \cdot |y_n|$ goes to $0$ as $n$ goes to infinity.
I thought the argument of adding and subtracting $xy_n$ was elegant, but unintuitive when I first saw it. I didn't memorize that proof, but I did pay a lot of attention to that part. And, as a result, I am able to recollect the whole proof, and that technique of splitting up $x_ny_n - xy$ I have seen several times in other places in real analysis, in particular $L^p$ spaces.