The case was brought by an interracial couple, Mildred and Richard Loving, who had gotten married in Washington DC in 1958 and returned to Virginia to live. They were arrested, tried and convicted in the town of Bowling Green, VA for violating the white supremacist 1924 Racial Integrity Act.
They were sentenced to a year in jail, but the sentence was suspended under the condition that the Lovings leave Virginia and not return for 30 years not even to visit family.
The Lovings moved to Washington DC, and Mildred Loving wrote a letter to then Attorney General Robert F. Kennedy appealing for help in their case.
That letter was forwarded to the ACLU, whose lawyers subsequently filed the legal case that went to the Supreme Court. On June 12, 1967 the SCOTUS unanimously ruled that Virginia's laws against interracial marriage violated the 14th Amendment and were unconstitutional.
That ruling overturned similar laws still on the books in 16 states, and led to a surge of interracial marriages. In addition, the precedent set in the Loving v Virginia case was cited in several cases holding that restrictions on same sex marriage in the United States was unconstitutional, including the 2015 Obergefell v Hodges SCOTUS decision.
"I believe all Americans, no matter their race, no matter their sex, no matter their sexual orientation, should have that same freedom to marry. Government has no business imposing some people’s religious beliefs over others. Especially if it denies people’s civil rights."
Richard Loving died in 1975 and Mildred Loving died in 2008, so they didn't get to see that legal precedent they set by standing up against injustice being used to strike down injustice against another group of marginalized Americans.
So while this case wasn't exclusively about love conquering all as the movie Loving would have you believe, this case at its core was about striking down white supremacist oppression and an unjust law.
At the same time, to borrow a line from Battlestar Galactica, you cannot declare war on love.