Why do we look at morphisms?

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.

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.