If you extend your map out a little bit further, you will see that the Brooks Range is just a continuation of the Rocky Mountains into Alaska.
Source: freeworldmaps.net, ©Daniel Feher (explicitly permitted use here)
The Brooks Range was formed at the same time as the front ranges of the Rockies in Canada and the Lower 48, during the Laramide Orogeny which lasted from abut 80 million years ago to about 35 million years ago. This was caused by the subduction of oceanic plates (mainly the Farallon Plate but also the Kula Plate, and to a lesser extent, the Pacific Plate) beneath the west coast of North America.
The Farallon plate went under North America at a shallow angle; because of this, a large part of Western North America was uplifted, deep into the continent, with faults pushing ancient basement rocks up where the Front Ranges and the Brooks Range are today.
Pacific Ocean plates, 55 Ma. Source: Wikimedia Commons
The Kula plate is "all gone", and the Farallon Plate is mostly gone, except for a few remnants like the Juan de Fuca Plate. Because of the different direction that the Pacific Plate moves, subduction ialong the West Coast is mostly gone as the Pacific Plate slides past North America. The shallow subduction that created mountains deep in the interior is done. A subduction zone further west has created the Aleutian volcanic arc, but this is a much steeper subduction.