I have found Kenji Ueno's book Algebraic Geometry 1: From Algebraic Varieties to Schemes to be quite satisfying in introducing the basic theory of schemes. Well, to be fair, this is only the first in a series of three books on the subject by the same author. So this first volume basically just develops the definitions of an affine scheme first and then of a scheme in general by "pasting" together affine schemes. It does not go into cohomology and more advanced stuff, which is the subject of the other two books.
However, what I really like is that he motivates very carefully the passage from the definition of an affine algebraic variety as an irreducible algebraic set in an affine space $\mathbb{A}_{k}^{n}$ to the definition of an affine variety using schemes, which is where you are having some trouble. What he does is that he starts by doing some algebraic geometry in the classic sense, that is, over an algebraically closed field $k$, in the first chapter of the book.
Then the author proves that there is a correspondence between the points in an algebraic set $V$ and the maximal ideals of its associated coordinate ring $k[V]$, where a point $(a_1, \dots , a_n) \in V$ corresponds to the maximal ideal of $k[V]$ determined by the ideal $(x_1 - a_1 , \dots , x_n - a_n) \in k[x_1, \dots, x_n]$, that is, a correspondence between the points in $V$ and the "points" in the maximal spectrum $\text{max-Spec}(k[V])$ of the coordinate ring $k[V \, ]$.
Then Ueno goes on to define an affine algebraic variety as a pair $(V, k[V \, ] )$ where $V$ is an an algebraic set. But he then makes the argument that one can go a little bit further and consider the pair $( \text{max-Spec}(R), R )$ where $R$ is a $k$ algebra. But here Ueno arguments that if the original intention was to study the sets of solutions of polynomial equations, then where is the geometry and where are the equations hidden if an algebraic variety is defined as this pair $( \text{max-Spec}(R), R )$?
The interesting thing is that if the $k$ algebra $R$ is finitely generated over $k$ then
$$ R \simeq \frac{ k[x_1 , \dots , x_n] }{I}$$
so that as a consequence
$$ \text{max-Spec}(R) = V(I)$$
so that again you'll have some equations (this is all done and explained in detail in the book). So then the author (re)defines an algebraic variety over an algebraically closed field $k$ (remember that he is doing everything in the classic sense) as a pair $( \text{max-Spec}(R), R )$, where $R$ is a finitely generated $k$ algebra.
And then at the end of the first chapter the author motivates the need for a more general theory, for example having in mind the needs of number theory, because since everything was done in the context of an algebraically closed field, then the arguments don't work for the fields (and rings) of interest in number theory. In particular, it is noted how an extension of the definitions to include these cases would need to take into account not only the set of maximal ideals, but the set of all prime ideals.
Then chapter two develops first some properties of this set of prime ideals, or prime spectrum of a ring, making it into a topological space with the Zariski topology... and then defines the necessary things in order to be able to define an affine scheme and a scheme (I mean, the concepts of a sheaf of rings, a ringed space, etc).
It is not a short story of course, but again I prefer this type of approach at first, than having to deal with an unmotivated (and difficult) definition that strives for great generality but I have no idea of where it comes from and what is its purpose.
Note that the book that Arturo recommended is great also but it assumes you already know some algebraic geometry and its level is higher than Ueno's book.
You should take a look at it and see if you like it, the book has a fair amount of examples and some exercises interspersed within the text also. You'll have to study from other sources as well but I believe that this book does a pretty good job at motivating the abstract definitions.
I hope this helps at least a little.
Best Answer
I rephrase my comment for clarity, answering the OP’s comment as well.
$U$ isn’t integral, but we can find it has an “almost integral structure”.
Indeed, for each $x \in U$, $O_{X,x}$ is a domain. With a slight back-and-forth with commutative algebra, it implies that there is only one irreducible component of $U$ going through any given point.
So the (finitely many) irreducible components of $U$ are pairwise disjoint and closed. As a consequence, they are open. Therefore, $U$ is a disjoint reunion of integral open subschemes.