They are not inherent bonuses.
If so, then why does the table show ability modifiers that are higher than +5 after ability scores get in the 22+ range?
"Inherent" is a type of bonus. Inherent bonuses are not supposed to be able to exceed +5 by nature of the bonus (though some things in Pathfinder such as sorcerer bloodlines break this rule). Inherent modifiers are instantaneous and cannot be dispelled. They are most commonly gained through wish spells, though there are overpriced wondrous items that can grant them as well.
So then my players can have a bonus higher than +5 from just what their score is? The phrase "Each ability, after changes made because of race, has a modifier ranging from –5 to +5." on this page (http://www.d20pfsrd.com/basics-ability-scores/ability-scores) is confusing me.
Yes they can. The catch is that you cannot generate a score higher or lower than 1 and 20. For example, if you roll your abilities you get a number between 3 and 18. Core races may have +2s or -2s to their stats.
EDIT: And if you use point buy ("Purchase"), your ability scores range between 5 and 20 at 1st level.For example, if you roll an 18 and a 3 on a dwarf (+2 Con, -2 Cha) then you could get a 20 Constitution and a 1 Charisma. This is only at 1st level mind you. Your ability scores increase every 4 levels, and can increase via magic items, inherent modifiers, and other beneficial buffs.
EDIT 2: Expanded racial options may change this a bit. For example, goblins get +4 Dexterity, so a goblin can begin play with a 22 Dexterity at the most. Keep in mind that racial penalties cannot (or should not) ever reduce an ability score to less than 1, and sentient creatures should never have an Intelligence score less than 3 (3.x spelled out out that sentient creatures default to 3 Int if their Int penalty would bring them lower than 3 Int).
The "-5 to +5" bit only applies when you're first starting out. Later in the game (or possibly even from the start if you're using certain ARG races), ability scores can be higher and the modifiers will grow with them. Look at this chart on that same page.
Okay! Thank you all for your quick responses. I will adhere to that rule at the start of character creation and then ignore it for the rest, knowing that they are not inherent bonuses or modifiers.
You should ignore it at the start as well. Its assumed the maximum ability score at 1st level is 20, which is a +5 modifier. Thats true for all of the core races, who have at most a +2 racial modifier to an ability score. But its possible to be higher at 1st level. An orc can start with a strength of 22 (+6 modifier).