Confusion about thresholding for solving lasso problem

This is derived by doing a proximal gradient algorithm, where you at each step minimize a convex upper bound on the lasso problem. If the function you're minimizing looks like f(x) + lambda |x|_1, where f is smooth (has a lipschitz continuous gradient), you can upper bound this function by

f(x) + lambda |x|\_1 <= f(x\_0) + grad f(x\_0)^T(x - x\_0) + C ||x - x\_0||^2 + lambda |x|\_1.

Minimizing this upper bound leads to a gradient-and-thresholding algorithm, which can also be implemented as doing a gradient step with learning rate 1/C and then applying a proximal operator. See how to derive the tresholding operation in this previous answer.