Why are some genes dominant over others? What is the mechanism behind it?
One version or allele of the gene for eye color may encode instructions to make the brown pigment, while the blue allele makes no pigment. Alternatively, it is possible, logically speaking, that the blue allele blocks the productions of brown pigment, but this is not the case.

Now, brown vs blue is not a case of one gene and two alleles. After producing pigment, it must be transported, deposited and such. Many processes are involved. So, the pigment deposition process may be functional or not, but as above, it is often found that a non-functional version is simply non-functioning and not actively inhibiting the process. There are numerous cases where an aberrant form of a protein, encoded by a variant allele, acts as an active inhibitor of a given biological or biochemical process.

The key to your question as written is blue eye color is a lack of pigmentation.