Springer developed his resolution of singularities for the unipotent (or nilpotent) variety of a semsimple algebraic group (or its Lie algebra) as part of a bigger program to construct Weyl group representations in a new way by making the Weyl group act on cohomology groups of fibers of the resolution. This is a multifaceted subject by now and can be approached in somewhat different ways. As Mike indicates in his answer, Ginzburg and others provide at least a partial pathway (mainly in characteristic 0) using modern algebraic geometry and emphasizing the best-behaved example $SL_n$. Some of the machinery used is necessarily rather sophisticated, as in Springer's original papers.

Papers by Steinberg and Spaltenstein developed a lot of detail about the Springer resolution and its fibers, including dimension formulas. Steinberg's 1974 Springer Lecture Notes 366 volume is based on Vinay Deodhar's write-up of Steinberg's Tata lectures covering some of this material. In my 1995 AMS monograph Conjugacy Classes in Semisimple Algebraic Groups I gave a more comprehensive treatment in the framework of the Borel-Chevalley structure theory of semisimple (or reductive) groups over arbitrary algebraically closed fields. See especially Chapter 6 for the Springer resolution in this setting, as well as the brief survey in Chapter 9 of Springer's construction of Weyl group representations (which Ginzburg outlines in his Chapter 3).

The Springer resolution taken in isolation is well enough presented in these sources, though in different styles and with different assumptions on the field. But there is no standard textbook treatment of the related Weyl group story, which is much harder to approach straightforwardly. Carter's 1985 book Finite Groups of Lie Type does contain a lot of concrete details in special cases, however. Which source to consult depends a lot on what you want to do next.