(cont.)>many problems involving varieties over non-algebraically closed fields, especially imperfect fields.
This is more of a general principle: it is often very useful to look at the solutions of a system of equations over a field in extensions of that field or more generally algebras over that field.
If you want to solve a problem over a non-algebraically closed field, a basic idea is to extend to the algebraic closure, solve the analogous problem, and see if you can transfer the result to your original field.
This method is called base change + descent. Sometimes, descent is easy and does not require scheme theory (for example, you don't need scheme theory to prove that two real matrices that are similar over C are similar over R: exercise), but sometimes you do need something stronger, like faithfully flat descent, to prove that a property over descends to k.
For that reason, people try as hard as possible to make constructions that are preserved by base change and faithfully flat descent.
For example, it has led algebraic group theorists to adapt group-theoretic concepts (normalizers, centralizers, kernels, etc.) to make them invariant under base change and descent. It makes them harder to compute, but very useful for proofs.
Algebraic group theorists use this "base change and descent" very often. An example of this is the classification of linear algebraic groups over an arbitrary field (still in progress). It relies heavily on the results over an algebraically closed field (achieved by Chevalley, Borel, Tits etc.) and scheme theory.