For the "why" you should use the tetrahedron method when computing the density of states, check out this paper on exactly that topic! In short, other approaches (e.g. Gaussian smearing) will occasionally obscure (and artificially introduce) some features of the density of states.
As for how the method works, quoting from the paper directly about the tetrahedron method:
"The tetrahedron method... divides the Brillouin zone into tetrahedra, calculates the eigenenergies at the corners of each tetrahedron, and linearly interpolates the eigenenergies inside of each tetrahedron to perform the integration. The tetrahedron method is reminiscent of the trapezoidal rule for approximating the integral of single-variable functions."