At its core, genes & DNA are blueprints for the assembly of proteins. Some proteins (often catalysts) are significant enough that their presence/absence will have an effect on the phenotype. Whether you have 1 or 2 working copies of those blueprints, they're enough to cause this dominance effect.

Keep in mind that most genetics is a bit more involved than 1 pair of genes broken down into binary pairs, but that's the brief mechanism