The Inbox menu contains 2 tabs: Inbox and News. You can access it directly from your Hangar.
How to access your Inbox
Inbox is the tab where you can find all the unclaimed rewards and gifts. They are stored there for 60 days and then automatically added to your balance. You can claim all rewards immediately by tapping the button "Claim Rewards" in the bottom right corner.
In-game News
In the News tab, you can learn about the latest updates and find details about new Mechs, Weapons, Events, etc. To mark all the news as read, tap the button "Mark All Read" in the bottom right corner.