Since this is directed to undergraduate students a more specific question is how does one acquire mathematical understanding by taking classes? But that does not mean that classes are the only way to learn something. In fact, they often are a bad way! You learn by doing. For example, it's questionable that we should have programming classes at all, most people learn programming much more quickly and enjoyably by picking a programming problem they are interested in and care about, and solving it. In particular, when you are no longer a student you will have acquired the skills necessary to learn anything you like by reading and communicating with peers and experts. That's a much more exciting way to learn than taking classes!
George Dantzig (himself the son of a mathematician) received a Bachelor’s degree from University of Maryland in 1936 and a Master’s from the University of Michigan in 1937 before completing his Doctorate (interrupted by World War II ) at UC Berkeley in 1946. He later worked for the Air Force, took a position with the RAND Corporation as a research mathematician in 1952, became professor of operations research at Berkeley in 1960, and joined the faculty of Stanford University in 1966, where he taught and published as a professor of operations research until the 1990s. In 1975, Dr. Dantzig was awarded the National Medal of Science by President Gerald Ford.
In this respect, those content standards which set an expectation of understanding are potential "points of intersection" between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.