I'm trying to learn category and set theory, and homomorphisms are mentioned right off the bat, and I'm having trouble understanding them. I know this question has been posited time after time on this website, but the answers and questions usually already have some weight of knowledge on this subject I haven't even attained yet. I'm completely new to anything related to this kind of abstract mathematics, and as such even the most basic forms of jargon on this subject is something I struggle with.
Basic examples homomorphisms I come across seem oddly unremarkably complex in what they're essentially stating but I can't piece together what it all means. Let me put this into an example. Here's an excerpt from Wikipedia on an example of a homomorphism:
Here's what I'm interpreting:
The set of real numbers is a ring. The set of all 2x2 matrices is a ring.
We're defining a function that is in a matrix and thus in the set of all 2x2 matrices but it uses elements of the set of real numbers as inputs. Thus, it is a function using two sets, both rings.
If a set preserves addition and multiplication in its operations, it can be called a ring.
Since, in this operation, which takes the set of all 2x2 matrices and the set of all natural numbers, applies $f(r)$ and maps it to a new set (let's say $\gamma$), and $f(r+s) = f(r)+f(s)$, this still preserves addition and multiplication (although I'm only showing the addition bit here but it's shown in the graphic).
Since this mapped elements of two rings to create another ring, this function is a homomorphism of rings.
Since this function preserves the algebraic structures of the sets being used, it is a homomorphism.
I assume some bits of my interpretation are incorrect, and I beseech any logical fallacies to be fixed for my understanding.
If this is true, then what implies that the elements of set $\gamma$ have preservation of addition and multiplication?
Finally, how can I relate this to my general understanding of morphisms if most of what I'm saying is true? In other words: in a general sense then, what are morphisms? What happened if preservation of addition and multiplication wasn't preserved? I've honestly never seen that happen before in my experience.