Me and a few of my group members came to a disagreement about when racial stat bonuses apply.

I believe that your racial ability score increases are only applied once at character creation. However, they argue that you continue to add those bonuses at every ability score increase as you level up. For instance, humans would get plus 1 to everything at ability score increases or half orcs would get a plus 2 to strength and a plus 1 to constitution at ability score increases.

I honestly think that isn't correct because it seems highly unbalanced. How does it actually work?

Please give answers with corresponding text from the PHB so I can see if it covers this question already.

Racial ability score adjustments are only added once at character creation.

The Player's Handbook is explicit about how ability scores improve. The ability score improvement class feature says:

When you reach 4th level, and again at 8th, 12th, 16th and 19th level, you can increase one ability score of your choice by 2, or you can increase two ability scores of your choice by 1...
[PHB, page 114]

There is no mention of adding any benefits from your race. Additionally, there is nothing in the racial descriptions that indicates you should add their racial bonuses when you increase ability scores.

You only apply the Ability Score Increase racial trait once

While it isn't explicitly stated that the racial trait isn't added throughout a characters levelling, if a specific feature intended you to improve every level it would be mention such as in the case of the Tough feat (emphasis mine):

Your hit point maximum increases by an amount equal to twice your level when you gain this feat. Whenever you gain a level thereafter, your hit point maximum increases by an additional 2 hit points.

All of the general rules for increasing features at level up appear in the section of the Basic Rules or Player's Handbook called Beyond 1st Level. This section discusses how class features, new hit dice, and proficiency bonus increases are awarded as you level up. Nowhere in this section are racial traits mentioned.

Further Evidence that the Ability Score Increase Racial Traits are a One-Time Bonus

PHB 11 (empahasis mine)

Your race also increases one or more of your ability scores, which you determine in step 3. Note these increases and remember to apply them later.

The rules state your race will increase your ability scores that are determined in step 3. It says nothing about increasing them with the standard occasional level increase of +2.

Later in step 3... PHB 13

Now take your six numbers and write each number beside one of your character’s six abilities to assign scores to Strength, Dexterity, Constitution, Intelligence, Wisdom, and Charisma. Afterward, make any changes to your ability scores as a result of your race choice.

After you assign your ability scores, you adjust them based on your racial traits. Again, no indication of changing them at any other time.

PHB 15 (Beyond 1st Level)

When your character gains a level, his or her class often grants additional features, as detailed in the class description. Some of these features allow you to increase your ability scores, either increasing two scores by 1 each or increasing one score by 2. You can’t increase an ability score above 20. In addition, every character’s proficiency bonus increases at certain levels.

Only class is referenced as granting additional features with leveling up. There is no reference to racial traits in connection with going up a level. Aside from Dwarven Toughness I can't think of any racial trait that increases with level, though I may be overlooking something esoteric.

PHB 20 (emphasis in original text)

Ability Score Increase. Your Wisdom score increases by 1.

Dwarven Toughness. Your hit point maximum increases by 1, and it increases by 1 every time you gain a level.

These Hill Dwarf traits provide the clearest insight into the RAW. The first trait, Ability Score Increase states your wisdom score increases by 1. This means it increase one time at creation, by 1 point. The second trait, Dwarven Toughness, is clearly intended to provide an increase with every level and this is clearly stated. Your hit point maximum increases by 1 every time you gain a level. Every racial ability increase is stated the same as this one, and makes no mention of any increase with level.

As stated in a previous answer, the PHB makes no reference to racial ability score increases in any of the Ability Score Improvement descriptions for any of the classes (PHB 67, Druid, for example)

The complete omission of the racial ability score improvement trait at all points in the PHB that discuss Ability Score Improvement with level make it quite clear that racial traits do not provide ongoing improvements to ability scores. They are a one time bonus at character creation.