You roll the die at the time the effect resolves, and perform the corresponding effect.[1]
If resolving "1 or 2: Immediately after this effect resolves, you can Normal Summon 1 monster", Normal Summon 1 monster from your handface-up. (If the monster is Level 5 or higher, it is still necessary to Tribute monsters as usual for a Tribute Summon.)[1]
If resolving "3 or 4: You can Special Summon 1 monster from your GY", choose and Special Summon 1 monster which can be Special Summoned from your Graveyard.[1]
If resolving "5 or 6: You can Special Summon 1 Level 5 or higher monster from your hand", choose and Special Summon 1 Level 5 or higher monster from your hand.[1]
Q&A Database Rulings
Q: "Vanity's Emptiness" is face-up in my opponent’s Spell & Trap Zone, and its effect that prevents players from Special Summoning is applying. In this situation, can I activate "Summon Dice"?
A: In this scenario, even if the effect of "Vanity's Emptiness" is applying, you can activate "Summon Dice". If the die roll result is 1 or 2, you can apply the effect normally, but if the result is 3, 4, 5, or 6, the effect of Special Summoning a monster cannot be applied.[2][3]
Q: I applied the effect of "Double Summon" this turn and already Normal Summoned/Set twice. In this situation, if I activate "Summon Dice" and roll a 1 or 2, can I Normal Summon/Set a monster with its effect?
A: In this scenario, you can still activate "Summon Dice" even if you have already Normal Summoned/Set twice by applying the effect of "Double Summon". As a result, if you roll a 1 or 2, you can still Normal Summon/Set with the effect of "Summon Dice".[4][3]