Because interracial marriage and intimate relationships were still illegal in several US 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. (Traveling there from their Virginia residence for the sake of marriage, and then back to Virginia, was not legal according to Section 20-58 of the Virginia Code.)
The prominent display of the certificate could have been an act of pride, defiance, or assertion. They did not "have to" display it that way by any Virginia law, as the marriage itself was not considered legal by Virginia authorities anyway.
Actually, there is a Wikipedia article about that couple's court case that eventually led to overturning the last of those laws as the Supreme Court considered such laws as unconstitutional.