Understanding the intuition behind math

I like this question a lot and I think that it's an important one. So here goes a (necessarily incomplete) attempt at answering such a broad and personal question.

First, "motivation" and "understanding for the essence" can mean very different things. There is of course physical motivation and intuition, and that probably applies most immediately to the Calculus III course that you are talking about. E.g. for the concept of derivatives of vector valued functions, you can think of the vector valued function of time that gives the position of an object as a vector. Of course, its derivative with respect to time will be the velocity (also a vector, since it described the speed and the direction of the movement) and the second derivative will be the acceleration. A good course in such an applicable subject will not just ask question like "compute the derivative of such and such a function", but will actually confront the student with real life examples.

But there is also intuition for less physical and more platonic concepts, such as that of a group, or of a prime number. Again, examples help. Also, you should always try to ask yourself the question "Could I have invented this?". If you see a new definition, ask yourself "What concrete problem might have prompted someone to define such a thing?". If you see a new result, ask yourself "Why was this to be expected, why would it be at least a reasonable conjecture?". Then try to convert your intuition into a proof. When you see a proof, ask yourself "Why is this a natural approach to try? Could I have proven this?". I agree with you that knowing the historical development can be very helpful in this and you should invest time in researching it.

I would like to contradict you in your assertion that intuition, motivation and historical context are black magic secrets that mathematicians acquire and then keep to themselves. It is true of some books and some teachers. So, you just have to find the right books. For that, you could ask for a specific recommendation here, including the area you want to learn and the books you have looked at, together with the reason you found them deficient. Of course, you can also ask specific "intuition" type questions.

To learn to appreciate mathematics, it is important to think about mathematics in your "spare time". Go out into nature and think about what your lecturer just told you in the last lecture. Or just think about whatever you find interesting. Then come back home with specific questions and look them up or ask them here.

Finally, something that I preach my students all the time is that they should develop a critical approach to what they are taught: if I give them a definition, they should try to come up with as many examples as possible. If a state a theorem of the type "A implies B", they should go home and find an example that "B does not necessarily imply A". If they do find such an example, they should ask themselves what additional hypotheses they need to impose to get the converse. If they don't, they should come back to me and ask me "but you haven't told us the whole story. What about the converse?".

In short, don't expect your lecturers to tell you everything you need to know. You should expect to have to think, to investigate yourself, to ask questions, and, above all, to think about mathematics because you can't help it, rather than because you are told to. This is not something, most people are born with, it's something that you have to cultivate.

I can offer a few things that help me gain an intuition for new material, in addition to the more mathematically-related advice given by others:

Talk about the mathematics, either with other students, or with your professor during office hours. I've noticed that while lots of times lectures can be dry and fast-paced, visiting after hours and asking about the concepts underlying the material can be very rewarding. Talking with other students in the class can also expose the meaning as long as you find a group of other students who are also interested in learning the "why" behind the "how." Ask each other questions ("what do you get out of this theorem?") and challenge the material.

Find books in the subject you are learning that are a bridge between the typical undergraduate text book and the more advanced (graduate level) books and read them as a supplement to the course. For instance, Tom Apostol's Calculus (Vol. 1 & 2) is a great companion book for the calculus series, and will answer many of the mathematical "why" questions that the typical undergrad books leave out (plus, you'll be a leg up when you get to advanced calculus!).

Lastly, read lots of mathematics books that aren't textbooks. I started learning about mathematics through books like Euclid's Window, Fermat's Enigma, and Prime Obsession. Journey Through Genius was also a good one, as it runs through the motivation behind several important results in mathematics. Since then I have switched to reading biographies and autobiographies to get to know the mathematicians who have come before me, how they thought, how they lived, and how their lives are all intertwined (some good ones: The Man who Love Only Numbers (Paul Erdos), The Man who Knew Infinity (Ramanujan), I Want to be a Mathematician (Paul Halmos)). In my experience, learning this side of mathematics has been invaluable for my understanding of the discipline in general.

Great teachers, great books, and (more recently) some great bloggers like Terry Tao and Tim Gowers. There are, in fact, some books that do a great job at giving context, rationale and intuition; Silverman and Tate's "Rational Points on Elliptic Curves" is one example, and some of it should be accessible to a motivated high-schooler. You should definitely read "Mathematics: A Very Short Introduction" by Gowers, and maybe browse the Princeton Companion to Mathematics from time to time.

Also, study physics.