Successfully is just a piece of new terminology, borrowed from YGO, to simplify casting and resolving. If it was countered or exiled or otherwise removed while on the stack, it was not successfully cast.
Soil's Bounty
Sorcery {C}
Put a 1/1 green Saproling creature token onto the battlefield.
Reverb - If you have successfully cast another spell this turn, put two 1/1 green Saproling creature tokens onto the battlefield instead.
Charmpyre Goliath
Creature - Giant {C}
Reverb - If you have successfully cast another spell this turn, Charmpyre Goliath enters the battlefield with two +1/+1 counters on it.
[3/3]
Voice of Thunder
Instant {U}
Voice of Thunder deals 2 damage to target creature or player.
Reverb - If you have successfully cast another spell this turn, Voice of Thunder deals 2 damage to target creature or player and 1 damage to another target creature or player instead.
Soil's Bounty
Sorcery {C}
Put a 1/1 green Saproling creature token onto the battlefield.
Reverb - If you have successfully cast another spell this turn, put two 1/1 green Saproling creature tokens onto the battlefield instead.
Charmpyre Goliath
Creature - Giant {C}
Reverb - If you have successfully cast another spell this turn, Charmpyre Goliath enters the battlefield with two +1/+1 counters on it.
[3/3]
Voice of Thunder
Instant {U}
Voice of Thunder deals 2 damage to target creature or player.
Reverb - If you have successfully cast another spell this turn, Voice of Thunder deals 2 damage to target creature or player and 1 damage to another target creature or player instead.