#You simply add the roll and the bonus together
Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target. Magic weapons, special abilities, and other factors can grant a bonus to damage.
—Damage Rolls, Player's Handbook, pg. 196
There's no language indicating that the damage must not exceed the maximum value of the die or dice, which is what your alternate interpretation seems to be based on. It simply instructs you to add the roll and the modifier together.
So if you were to roll a 1d8+3, rolled a natural 6 on the die, the total damage would be 6 + 3 = 9, with no capping of the damage due to exceeding the value that can appear on the die.