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.
I'm going to give some reason I don't think I can find the whole sets of reasons.
From here I'm going to refer to maps that preserve structure as morphisms.

1. Studying morphisms from one strucure always carry out some information about the structure itself, for instance looking at the group of automorphisms of two structures can helps us in saying if two structures are not isomorphic, isomorphic structures must have the some automorphisms group. In some cases, almost all, morphisms contain all the information about the structure: for instance the space $\mathbf{Top}(\bullet,X)$, of continuous function from the point space to the space $X$, is homemorphic to the space $X$, similarly the group $\mathbf{Grp}(\mathbb Z,G)$ of the homomorphisms from the group $\mathbb Z$ to the group $G$ with the componentwise product, is isomorphic to the group G and so on.
Other more interesting examples coming from to topology are loop spaces, foundamental groups, homology and cohomology groups (for instance singular homology which deals essentially with continuous maps from simplices in the space we want to study), but we can go on.
2. Often to study a given structure it is useful having various presentation, i.e. various example structures isomorphic to the starting structure in which we can made some calculations, that's so important because in such structures it's easier prove some facts about the structure itself. This is what we do when dealing with linear representation of groups or algebras, or group actions, or topological actions, or representations of vector spaces in $\mathbb K^\alpha$ for some ordinal $\alpha$, through a choice of a basis in the vector space (here $\mathbb K$ is the field subsumed).
3. Many properties relating structures can be expressed in a really useful way via universal properties i.e. the categorical language, which is the language of morphisms. This is really usefull for lot of reason, first of all it allows to see deeper unity in construction build up in different families of structures (i.e. different categories), helps to see how translate result from one environment (category) to another one, allows to generalize some result known in a particular class of structures to other similar, suggests the right definition for new concept in new environment (personally I don't know how to define the product between quasi-projective algebraic varieties without the morphism(category)-theoretic language).

Right now I'm not seeing other reasons, but I reserve to myself the right to add some stuff later.