The point to understand is that here you are not dealing with the colour adjectives but with the nouns of the colours, and that the indefinite article that is usually necessary (le rouge, le blanc, le bleu), is missing ; common nouns in French take the definite article and are not zero article nouns (as in English (red, white, blue)) ; it remains to know why they are missing.
The reason is that an enumeration is being made and it is often possible, as in English, to dispense with the articles in enumerations. Examples such as the following abound.
- La rame passa par dessus bord et il n'y pu rien.
- Rame et rameur passèrent par dessus bord et bientôt ne flottait plus que la rame.
- Dans un couvert normal il faut mettre fourchette, couteau et verre près de l'assiette.
Note that you could also write the sentence as follows ;
- Les couleurs du drapeau américain sont le rouge, le blanc et le bleu.
This could be deemed a preferable style (see @ Stéphane Gimenez 's comment).