I don't get how culture and language would be promoted via independence. Was it that the English Canadians were refusing to let the French language be taught in schools? How were they standing in the way of French culture?
The movement has since died down. Did the French Canadians achieve something meaningful?
There's always more to be said, but in the meantime you can find more info in this post from u/enygma9753 about French Canada's linguistic and cultural struggles amid a majority English-speaking North American culture.
It's an answer that requires much context and appreciation of Quebec's history from colonial times, which you will find in the link above.
When Quebec became part of the British Empire after the 1759 Conquest and formalized in the 1763 peace treaty ending the Seven Years War, its population was less than 80,000 mostly French Catholic settlers. In contrast, the Thirteen Colonies had a population of 1M+, mostly English and Protestant. This would swell to 2.5M by 1775.
Anxiety and fear of cultural/linguistic assimilation has always been an existential threat in French Canada, to this day.
Colonial and later federal officials implemented some protections for language and culture, from the 1774 Quebec Act to the guarantees in the 1982 Constitution and Charter of Rights.
For much of the colonial and pre-confederation period, they had been governed by anglophones and would only see more equitable representation during the reforming years of the 1840's when Lower Canada/Quebec received its own elected assembly, a response to the 1837-8 rebellion (which in Quebec was less about culture and more about political inequity.)
There's a concept in Quebec known as "maitres chez nous" (masters in our own house). Much of its politics is stemmed in this notion that Quebec cannot rely on English Canada alone to protect its interests and Quebec must protect and defend its language and culture itself. This manifests itself in many forms and both pro-federalist and separatist governments in Quebec have agitated and, at times, won special powers from the Canadian government to protect French language and culture. Not all Quebecois are separatists who see independence as the only recourse, but most would support increased powers and laws to protect their language and culture.
Successive federal governments' perceived failures on this front led to the 1976 election of a separatist government in Quebec and the 1980 referendum on independence, which was defeated. Further constitutional wrangling ensued in the '90s and dissatisfaction with this process led to the election of another separatist government. A divisive 1995 referendum was defeated by the narrowest of margins.
Support for separation has ebbed since then, but it still exists. Canada passed the Clarity Act in 2000, which outlines terms for negotiation if a province decided to secede from the federation. By law it would require more than a 50% + 1 majority vote in a referendum to trigger such negotiations, and only the House of Commons can determine if the vote had a sufficient majority and if the referendum question was clear enough. The separatist government at the time passed Bill 99, which asserts that Quebec can separate unilaterally with only a 50% +1 majority in a future referendum.
In Canada, Quebec has much clout as the second most-populous province after Ontario. Within North America however, their francophone population is dwarfed by 350M+ anglophones in America and Canada. In this context, they feel their language and culture is at risk or under threat. Quebec's concerns over assimilation have been prevalent in French Canada since Wolfe's victory in 1759.