I'm a graduate student of mathematics starting to study algebraic geometry with a focus on toric varieties (along Cox, Little, Schenck). From what I have learned so far, I can grasp that toric varieties form a subclass of algebraic varieties that can be studied using the combinatorics of polyhedral fans, where properties are easier to compute than in the general setting of algebraic varieties.
To this point, I have treated the subject rather abstractly and haven't seen any applications of the theory.
I would love to hear about some applications of toric geometry, to get a better understanding and probably some motivation. Where and why do toric varieties appear?
As Potato said in the comments, toric varieties are extremely easy to compute with compared with a general variety. As soon as the defining equations are not binomial, most algorithms are extremely slow. But if the variety is defined by binomials (i.e. it is a toric variety), then we can use lots of combinatorial machinery to compute. For example, for primary decomposition, there's an algorithm by Eisenbud/Sturmfels that is extremely fast, and works with binomial ideals.
It is written a book about algebraic statistics, whose algorithms mainly assume the equations are toric.
It also appearantly has applications in mirror symmetry in physics, see Martinez-Garcia Toric varieties in a nutshell for more details (page 32 and onwards). Basically, there's some duality going on because of the polytopes, some desingularization, and some deformation theory. I don't know the details, just the buzzwords.