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: 
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.
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.