This page of our Borderlands 3 game guide explains how to collect your mail after you have been notified in the game that you have received a new message or package. It's easy to do this, but accessing the mail menu is not very intuitive.
New mail messages appear in the upper left corner of the screen. You can receive the message after you have completed one of the hunts ordered by Sir Hammelock or completed a challenge prepared by a weapon manufacturer.
Pause the game and select Social from the list. You have to do this during the game. The check mail option will not be available if you have finished a gameplay session and returned to the main menu.
Check the Mail tab. You can only receive a text message or an additional reward package (example is shown in the picture). In the latter case it is necessary to select Accept to move the item to your inventory.