I would like to understand why the sum and product of algebraic integers are algebraic integers.
For algebraic numbers (not integers) there is the wonderful website https://www.dpmms.cam.ac.uk/~wtg10/galois.html which uses only basic linear algebra. A short version of that is in this MSE-answer: https://math.stackexchange.com/a/155153/564656
My Question:
Can this method or something similar be used for algebraic integers?
Yes, the exact same proof (at least, the brief MSE version) shows that the sum/product of algebraic integers is algebraic.
As in the linked post, take $V = F[x,y]/(p(x),q(y))$. Verify that because $p$ is monic, the matrix of the operator $\alpha(x,y) \mapsto x\,\alpha(x,y)$ has integer coefficients. The same holds for $\alpha(x,y) \mapsto y\,\alpha(x,y)$.
Now, the sum/product of matrices with integer coefficients is also matrix with integer coefficients. So, the matrices associated with $x + y,xy$ have integer coefficients. By the Cayley Hamilton theorem, $x + y$ and $xy$ therefore satisfy a monic polynomial with integer coefficients.