To answer the question in title of the posting (here I am rephrasing what I learnt from philosophical writings by several great mathematicians; Vladimir Arnold and Andre Weil are two names that come to mind, but there surely are others who said something similar, although I can't give you a reference now): because mathematics is discovered in one way and written in a very different way. A mathematical theory may start with a general picture, vague and beautiful, and intriguing. Then it gradually begins to take shape and turn into definitions, lemmas, theorems and such. It may also start with a trivial example, but when one tries to understand what exactly happens in this example, one comes up with definitions, lemmas, theorems and such. But whichever way it starts, when one writes it down, however, only definitions and lemmas remain and the general picture is gone, and the example it all started with is banned to page 489 (or something like that). Why does this happen? This is the real question, more difficult than the original one, but for now let me concentrate on the practical aspects: what can be done about it?
Here is an answer that I found works for myself: try to study a mathematical theory the way it is discovered. Try to find someone who understands the general picture and talk to that person for some time. Try to get them to explain the general picture to you and to go through the first non-trivial example. Then you can spend weeks and even months struggling through the "Elements of XXX", but as you do that you'll find that this conversation you had was incredibly helpful. Even if you don't understand anything much during this conversation, later at some point you'll realize that it all fits into place and then you'll say "aha!". Unfortunately, books and papers aren't nearly as good. For some reason there are many people who explain things wonderfully in a conversation, but nevertheless feel obliged to produce a dreadfully tedious text when they write one. No names shall be named.
Here is another thought: when one is an undergraduate or a beginning graduate student, one usually doesn't yet have a picture of the world and as a result, one is able to learn any theory, no questions asked. Especially when it comes to preparing for an exam. This precious little time should be used to one's advantage. This is an opportunity to learn several languages (or points of view), which can be very helpful whatever one does in the future.
