Soft question. Why do concrete problems motivate abstract theory? From what I learnt, I found that mathematicians develop abstract theory often in order to solve concrete, classical problems. For example, I read that ideals were introduced to solve problems in number theory, and that functional analysis was developed to capture the essence of many classical problems.
This is the opposite to the order in studying--we learn concrete examples in order to better understand abstract concepts. Also, going abstract from a particular problem means that we lose information. Then why does abstract theory help mathematicians tackle concrete problems?
 A: In general, when you try to solve a concrete problem, you look for an isomorphism between the reality and some abstract idea. Abstract idea are easier to be described, and working on them is often simpler than working directly on the real problem.
A: The prototypical example is Fermat's last theorem. The problem is so simple you could explain it to someone in grade school, but it was so difficult to solve that it led to mathematicians to further develop the theory of abstract ring theory in the quest for its solution. 
As for why does abstract theory help one tackle concrete problems I'd say the first example at the most basic level would be arithmetic. Humans needed to count things for many practical aspects of every day life. When people first started counting things they did it one-by-one: $1, 2, 3, \ldots$. But if you have a corn field with 160 rows of corn trees with 200 trees in each row it may take you a while to count how many corn trees you have if you do it simply one-by-one. But once the abstract structure of the natural numbers is established and the simple rules of arithmetic operations on this structure are developed then the counting problem becomes immediate: you have $160\times 200=32,000$ corn trees. So the abstract structure is motivated by the need to solve concrete problems. Then you focus on just studying that structure so that later on the concrete problems become trivial. Or maybe you just think the structure is beautiful by itself and study it for its own sake, then people call you a "pure mathematician".
