I have a druid ready to become a Verdant Master. I understand everything well enough except the "+1 level of existing class" found under spells per Day/Spells known chart. What does that mean to me?
It means that your character's spells per day and spells known continue to increase on the same level schedule as one of the classes they already took that qualifies for the prestige class, as though each level of the prestige class was a level of that class. It's only for the listed purposes, though: you don't automatically get e.g. Celestial Mount upgrades as a Paladin if you take a PrC that advances spellcasting, even though Paladin spellcasting would be advanced.
In this case, Druid 19/Verdant Master 1 has the same spells per day as Druid 20, and so forth; spells known is irrelevant.
+1 Level of Existing class states:
Spells per Day/Spells Known: At each verdant lord level, the character gains new spells per day (and spells known, if applicable) as if he had also gained a level in a spellcasting class to which he belonged before adding the prestige class. He does not, however, gain any other benefit a character of that class would have gained (additional wild shape options, metamagic or item creation feats, or the like). If the character had more than one spellcasting class before becoming a verdant lord, the player must decide to which class to add each verdant lord level for determining spells per day and spells known.
What it means to you, is his spells he could cast everyday and the caster level of those spells would be affected. Nothing else would be.
+1 Level of Existing Class
Caster Level goes up for a spellcasting class that you possessed prior to gaining that level of prestige class. Caster Level in this case means: Caster Level checks, Spells Per Day, and Spells Known. (As mentioned above, only applies to spellcasting.)
+1 Level of Existing Arcane Spellcasting Class
Same as above, for Arcane casting specifically (e.g. Bard, Sorcerer, Wizard).
+1 Level of Existing Divine Spellcasting Class
Same as above, for Divine casting specifically (e.g. Cleric, Druid, Paladin, Ranger).
+1 Level of Wizard Spellcasting Ability
Same as above, for Wizards only.
+1 Level of Prepared Arcane Spellcasting Class
Same as above, but for arcane casters that prepare spells ahead of time (e.g. Wizard).
+1 Level of Spontaneous Arcane Spellcasting Class
Same as above, but for arcane casters to don't prepare spells ahead of time (e.g. Bard).
+1Level of Lower-Level Exisitng Arcane Spellcasting Class
Same as above, but applies to an arcane casting class that is lower than the other.
+1 Level of Existing Invocation-Using Class
Same as above, but applies to an invocation using class (e.g. Warlock)
+1Level of Existing Class Features
Allows advancement of an existing class' entire class features (e.g. Paladin's Mount)
Note: Technically, if advancing wizard spellcasting using a prestige class's "+1 Level" feature, you do not gain new spells in your spellbook (or advance your familiar). That is a wizard feature, and not many (hardly any) prestige classes add new spells to your spellbook upon level advancement (or advance your familiar). However, I have not met a DM that enforces not giving new spells.