Speaking as an actual math teacher, this is a very important thing for students to try to come to grips with. Memorization and "learning without understanding" have a bad rep, but memorization is a tremendously valuable skill. Instead of thinking of "learning by memorization" as doing a disservice to learning math, consider that learning math might increase your capacity for memorization as a byproduct. This skill carries over to so many others, but I think its importance gets glossed over as a result of it being simple to look things up using the internet.
Another thing to think about: if you have something memorized (e.g., an identity, a theorem, a formula...), then you can think about it when you're walking around. When you have many of them memorized, you'll be able to think about how they relate to each other. The more you have memorized, the fewer stumbling blocks there will be to trip over when you're mulling over a problem or trying to figure something out.
There's a lot of apocrypha out there about how if you're learning math, you shouldn't try to memorize things. You should instead try to "pick it up". This elides two important details: 1) there's selection bias at work---many people who are extremely good at math have minds like steel traps, remembering many things after seeing them only a couple times, 2) the expectation is that you spend a significant amount of time drilling and working, to give yourself an opportunity to pick it up.
Now, why did his niece say this? When you're learning the basics like this, becoming "comfortable with formal manipulation" is really a matter of running a large number of experiments. You try this and that manipulation and get comfortable with them over time. You apply a formula in error and observe the results. You puzzle over what happened. Over time, you discover the logical framework which holds everything together.
Asking "why" at this point (when you're learning very basic math!) is like putting the cart before the horse. You would receive an answer that you would be in no position to understand. Hence, "be like a child": play around, make some mistakes, try to understand what's going on. As you grow, you will become more sophisticated in your approach and better able to shift the balance so that you can attempt to simultaneously understand more of what you're learning.