Question: I am reading some lecture notes and in one paragraph there is the following motivation: "The best way to study spaces with a structure is usually to look at the maps between them preserving structure (linear maps, continuous maps differentiable maps). An important special case is usually the functions to the ground field." 

Why is it a good idea to study a space with structure by looking at maps that preserve this structure? It seems to me as if one achieves not much by going from one "copy" of a structured space to another copy.

Answer: 
There is no short and simple answer, as has already been mentioned in the comments. It is a general change of perspective that has happened during the 20th century. I think if you had asked a mathematician around 1900 what math is all about, he/she would have said: "There are equations that we have to solve" (linear or polynomial equations, differential and integral equations etc.).

Then around 1950 you would have met more and more people saying "there are spaces with a certain structure and maps betweeen them". And today more and more people would add "...which together are called categories". 

It's essentially a shift towards a higher abstraction, towards studying Banach spaces instead of bunches of concrete spaces that happen to have an isomorphic Banach space structure, or studying an abstract group instead of a bunch of isomorphic representations etc.

I'm certain all of this will become clearer after a few years of study.
Studying the structure of an object is producing theorems about this object that gives you information about this object.
If you want to understand the structure of an object by studying how it maps to another objects, then you $\textbf{have}$ to retain your attention to the maps that preserve the structure ! You can't expect to obtain information about the structure if you are looking at maps that don't have anything to do with it. In this case, preserving the structure is the least you can ask to a map you are looking at.

The two important questions, are, I think : why do we study structures ? what can we expect from the study of the maps preserving these structures ?

Why do we study structures ?

The mathematicians work, I think, is to detect some structure $\textbf{hidden}$ in an object, then study the structure, and hope it will help to understand the object better.
Two examples : the observation that the set of the permutations of a set $X$ possesses a natural operation, the composition of maps, and that this operation has a certain number of well-known properties led the mathematicians to study the objects called groups. This observation has been one of the most important in the history of math. The second example is the field $\mathbb{C}$ constructed by adding some special element $i$ to $\mathbb{R}$. It is just an algebraic object, until you discover that its topology has very interesting topological properties, which turn out to be very useful when you want to prove that it is algebraically closed.

Why can we expect from the study of the maps preserving these structures ?

If you want to study the structure of an object that maps to another object about which you know a lot, then you can expect to $\textbf{transport}$ the information that you know, using these maps ! It happens all the time : if you have a one-to-one linear map between two vector spaces, you can deduce that the dimension of the first is smaller that the seconds. If you have a continuous surjective map from a connected space to a topological space, you can deduce that it is connected.