You first need to realize that Turing machine were invented before the first computer was ever built, and they were born as a mathematical model, an ideal construction.
The problem at the time was that there were certain natural classifications of functions on natural numbers, the recursive functions, and the Turing machine model helped to understand that partial recursive functions = computable functions.
Computable, at the time, meant that a human being was able to calculate them with the aid of pen and paper.

Nowadays, depending on the branch of computer science you want to study, either recursive functions or Turing machines are used as the default general model of 'computability', either to study specializations of that concepts (complexity) or to show that some class of functions cannot be computed (Turing jumps, oracles, etc.)
You can think of them as idealized computer, and they are a big deal in the sense that they are the cornerstone upon which all computer science is built, the same way 'the continuum' is a big deal for calculus.