As far as I can see, it just tells the DM to roll damage in the sequence of action outline. (Note that it does not say that the DM should roll player to hits.) And B/X is kind of all over the place with who should roll what.
For instance, it has the GM rolling surprise for both the players and monsters, but has the players roll their own initiative.
Bottom of B23:
>INITIATIVE: To determine initiative, each side rolls Id6 (the DM rolls for the monsters).
Example of play on B28:
>The DM rolls a 2 for the hobgoblins' initiative; Silverleaf rolls a 4 for the party, so the party has the initiative.
Also on B28:
>Morgan rolls a 12 and Fredrik rolls a 16 — both hit! The DM rolls 1d6 for arrow damage and 1d6 for axe damage.
Notice how it has players rolling to hit and the DM rolling damage. But later on the same page, it seems to contradict this, though it's hard to tell because they reference the character names for both player and character actions:
>Rebecca and Silverleaf both miss, but Morgan hits (with her sword). She rolls a 4 for damage.
The example of play on B59 has the DM making most of the rolls, but the players rolling initiative and the saving throw vs. poison that kills Black Dougal.
B60:
>The DM answers: "Looking down into the chasm, your character can estimate that he has a 98% chance of dying, no saving throw, if he jumps. If you decide your character jumps, roll percentage dice. A result of 99 or 00 will mean that your character lives, but any other result will mean that he will die in the attempt."
B60:
>To perform a difficult task (such as climbing up a rope or thinking of a forgotten clue), the player should roll the ability score or less on 1d20.
B9:
>If the player rolls the given number (or greater) on 2d6, then some of the undead monsters are Turned.
>If a cleric is successful at Turning some undead monsters, the player must roll 2d6 to find how many hit dice of undead monsters are Turned.