What are some interpretations of Von Neumann's quote? John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics. 
Did he mean that with experience and practice, one obtains understanding? 

One thing he might have been referring to is that in mathematics you often have to learn to apply a method without actually understanding what it is all about.
Take for example matrix multiplication. You could (and many students do) beat themselves up about why it is so "weird" in comparison to say multiplication of the reals. But it turns out that yes it has those weird properties because it is perfect for representing a linear transform, amongst other things.
In general I have found it counter productive to try and understand every aspect of something before moving on to the next thing. I just accept that that's the way it works, trust that one day it will have some sort of application, be useful or otherwise "make sense".
Note that the history of mathematics is full of branches of mathematics that didn't even have this sort of utility when they were initially created and explored, but have later turned out to be enormously important. Take for example Boolean algebra and knot theory.
Another important point is that mathematics is the study of abstract logical systems, including wholly invented ones. Therefore it can be pretty fruitless to understand some of the deeper meanings of a mathematical concept, because they might not even exist. Sure there might be deep connections or generalisations to other mathematical concepts, and applications might be found, but trying to say that the application is "the true form" of the mathematical concept is putting the cart before the horse.
