The motto works in two directions.
From liberty, diversity
Give people genuine freedom, and real diversity follows. Not the managed, top-down version of diversity that many Western governments promote, where certain differences are celebrated and others are quietly discouraged, but the kind that emerges on its own when nobody is forcing everyone into the same mold.
The difference matters. Forced multiculturalism is still force. It decides which identities count, which communities deserve recognition, and which traditions are worth preserving. Real liberty does none of that. It simply gets out of the way and lets people build communities around whatever they actually share: language, work, land, belief, tradition, or something else entirely. The variety that results is genuine because nobody designed it.
From diversity, liberty
The second direction is less obvious but just as important, and it speaks to something a lot of people feel without having words for it.
When a society flattens real differences in the name of unity or progress, many people end up feeling like strangers in their own country. Their community does not fit the national template. Their way of life is tolerated at best, quietly discouraged at worst. That feeling of alienation is not a personal failure. It is what happens when diversity is declared rather than allowed.
When communities are genuinely different and genuinely free to be so, the opposite happens. People can live in a place that actually reflects who they are, among others who share their values and their way of life. They do not have to constantly negotiate their identity against a dominant culture. That is what it actually feels like to be free: not just the absence of legal restrictions, but the presence of a community you chose and that chose you.
Real diversity makes that possible. And that is why diversity, when it is real, always leads back to liberty.
Live and let live is the conclusion both directions arrive at.