I feel that the explanation provided by u/orcris covers only the modern period, it doesn't explain the real reason why French still is the dominant language in Quebec. Pardon my English, French is my first language.

First, the French colony was conquered by the British in 1760, it was officialized by the Royal Proclamation of 1763 . The British now had a French colony to administer. The population was not large (around 60K), while the American colonies were much larger (about 2 millions).

The British tried to appease their new French citizens by enacting the Quebec Act in 1774. Among else, Catholicism was no longer restricted, civil servant no longer had to renounce Catholicism, French civil law was restored (save for criminal proceedings, were common law was, and still is, in use). More importantly, the Church could levy taxes and alms. The Catholic Church was a huge help to the British, it helped to reign in the majority French population.

Shortly after, the American colonies obtained their independence. Loyalists settled in the Province of Quebec. The loyalists were not numerous (about 70K), they mostly settled in the western, not populated, part of the Province of Quebec. in 1791, Quebec was divided in two with the creation of Upper and Lower Canada, now respectively Ontario and Quebec, so the loyalists could live under the British laws, customs and culture.. The two provinces were roughly equally populated.

Up to the modern birth of Canada,the signing of the Canadian Confederation in 1867, the two provinces were roughly equally populated. Upper Canada (Ontario) had 1.6 millions persons, while Lower Canada (Quebec) had 1.2 millions. The Anglophone communities were mostly found in Western Quebec, close to the US border and in the Anglophone elite of Montreal and Quebec City. French survived simply because the communities were tight knitted, profoundly religious and led by Canadian (which meant French-Canadian at the time) nationalists.

It's only when the Atlantic Provinces joined the confederation and when the Prairies provinces were created in 1905 that the weight of the French population in Canada began to (relatively) decline. Now we are around 7.5 millions of the 35 millions (24%). But 7 millions Francophones is still a large community, since we are primarily based in a single province. The policies explained by u/orcris only cover the recent period, from the second world war to now.

It's important to remember that the French language isn't limited to Québec or that all French speakers in North America are Québecois. There were 7 million French speakers in Canada as of the 2011 census - 6 million of which are in Quebec. However there are also a half million in Franco-Ontarians. On the east coast (although almost all in New Brunswick), there are around 300,000 Acadians who are culturally very different from Québec - with a very different history and their own dialect of French.

The remainder are spread over rest of the country, notably in places like St. Boniface in Winnipeg and in other enclaves.

Source: Statistics Canada 2011 Census

Bill 101. In provinces with higher percentages of English speakers, French was eradicated by the school system, which taught classes in English, and because English was the prestige language.

In the 1970s, French was on the way to dying out, since most employers heavily favored English and English speakers earned much more than French speakers. During the Quiet Revolution, which was a period of rapid liberalization and secularization, Quebec nationalism began to appear and Quebeckers began wanting equal treatment with anglophones. A nationalist Parti Qiébécois government was elected in 1976 and passed the Charter of the French Language, or Bill 101. This law required all products in Quebec to be translated into French, all students without parents from an anglophone province to be educated in French, and it outlawed discrimination against French speakers in businesses. Because of Bill 101, French has gone from being on the verge of dying in the 1950s to the dominant language in the province today.