Why do we look at morphisms? 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.  
 A: It is hard to answer this question because it is hard to say what mathematicians mean when they talk about structure. Various attempts to define structures and their ultimate failure are chronicled in the book Modern Algebra and the Rise of Mathematical Structures by Leo Corry.
Instead of trying to give an intrinsic definition of structure, we can look at what we understand by and what we do with structures in mathematics. Intuitively, structure is something deep that goes beyond mere surface properties of an object. Different mathematical objects are structurally the same, the differences are superficial. 
An indirect way of delinating structure is to define what it means to change the superficial properties of an object without actually changing the structure. As a simple example, take the process of adding up "objects". You can add up three apples and two apples by getting a bowl containing three apples and a bowl containing two apples. You "add" them by pouring the content of one bowl into the other bowl. Now apples are very concrete objects, but there is structure behind the process. There are numbers. You see that by replacing each apple in each bowl by an orange. Apparently, you can "add" oranges the same way you can add apples. When you replace each apple by an orange, you keep their number the same. And this abstraction process is essetially what we do when we use numbers in the real world. The concept of (counting) numbers is basically that there is some deep structure that keeps the same when we replace objects. Changing apples to oranges to stones to sheep to... are all transformations that do not change the underlying struture.    
An important step in geometry was the insight that one can define geometric structures by taking a class of transformations of a geometric object and declaring that structure is what is not changed by the transformations, it is what is invariant. This is essentially the gist of the Erlangen program in geometry, developed by Felix Klein and a huge step in the history of structural mathematics. This shows the use of replacing some object by an equivalent one. What we learn is that the process of replacing the objects doesn't change the structure and if we know all these admissible ways of replacing objects, we know the structure. 
So far, we have considered only reversible changes or transformations. But there are good reasons to allow for one-way transformations. The reason that they are useful is essentially the reason maps are useful. If we consider a country to be a map of itself, at least as useful as an actual 1:1-map of the country, we can replace the country by a simpler map that is sufficient for, say, a taxi driver. We can use the same map to draw an even simpler map that is merely good for getting from the railwaystation to the grand hotel. None of these processes are reversible, we cannot use the simple map to draw the bigger map without getting the needed additional information- or structure- from somewhere. So we can see these one-way transformations as ways to preserve parts of structure.  
Further Reading:
When is one thing equal to some other thing? by Barry Mazur gives a detailed motivation of the abstraction process underlying category theory and adds a lot of depth.
A Hundred Years of Numbers. An Historical Introduction to Measurement Theory 1887-1990 by José Diez (part 2 here) shows how the structural ideas matter when we want to formalize what it means to measure something in science. 
A: 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.


*

*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.

*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). 

*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.
A: 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.
A: 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.
