The Honor Hall was founded by Starfleet to provide particularly-accomplished Captains with additional resources.
Honor is a currency that can be earned by completing a number of daily personal and Fleet tasks, career-long achievements, and most notably by Dismissing recruited personnel to serve elsewhere in Starfleet.
As you accumulate Honor, you will eventually be able to consider redeeming it for valuable resources or other special authorizations, such as Honorable Citations used to upgrade a Crew member, or even recruitment of a unique Crew member.
Your Honor total is shown in yellow at the top of your screen when in the Crew Quarters, Time Portal, Fleet, and other locations. You can access the Honor Hall itself from its tab in the Time Portal.