I am hoping someone can break this section of code down and help me understand just how this functions.
const buildApi = (
  disk: Disk,
  newProp: {
     sector: Sector[]
  }
): ApiDetails => { }
You assign the function to the buildApi variable. The function takes two parameters, disk of type Disk, newProp of type (object) {sector: Sector[]} and the function is going to return something of type ApiDetails the => is the arrow in the arrow function, while everything inside the following { } would become the body of the function.
The above is equivalent of
var buildApi = function(disk, newProp) { }
or
/*
  This is only partially true, as this one would be hoisted, the above isn't
*/
function buildApi(disk, newProp) { }
In all cases, you would invoke the function by buildApi(disk, newProp);