FHWA Acts as a funder and regulator of highway planning, construction, and maintenance. From the funding perspective, they oversee dolling out gas tax revenue to states in the form of Federal Aid Highway program + some smaller programs. Money goes to state DOTs.
The Interstate system in each state is maintained and managed by the state DOTs; however FHWA has authority over them - approves any changes/revisions in use etc. They sign of on env. actions.
Each state can manage its interstate system differently e.g. tolls/HOV lanes etc but FHWA has the final say. They are also key to some innovation - Autonomous Vehicles and highway electrification (EV charging)