Because interracial marriage and intimate relationships were still [illegal in several US states](https://en.wikipedia.org/wiki/Anti-miscegenation_laws_in_the_United_States) at the time, including the state of Virgina, where the movie is taking place, at the time it *is* taking place. 

The couple got married in Washington D.C., where such marriage was legal. The prominent display of the certificate could have been an act of pride, defiance, or assertion.

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Actually, there is a [Wikipedia article](https://en.wikipedia.org/wiki/Loving_v._Virginia) about that couple's court case that eventually led to [repealing the last of those laws](https://en.wikipedia.org/wiki/Anti-miscegenation_laws_in_the_United_States#Laws_overturned_on_12_June_1967_by_Loving_v._Virginia).