I've worked with many Canadians and Quebeckers back in the day (I've also visited Canada/Quebec a few times), so here's my rather lengthy input:
First, you need to understand the historical context. It goes back hundreds of years when various European powers "discovered" the Americas. The French, Spanish and English (and to a lesser extent, the Dutch and others) all had North American colonies.
The British, however, were the only ones who actually put in some effort to colonize and invest in their American colonies. They invested more in infrastructure, promoted settlements to the colonies, had more farming and so on. On the other hand, the French had a very small population (60 000 settlers in New France vs 2 million in the English clonies). So despite the vastness of the French territory, it was sparsely populated and difficult to defend. The French got most of their income from fur trading, which at the time was worth decent money as beaver hats were all the rage in Europe. Probably the equivalent of an iPhone 6 today, everyone just had to have one and be seen with one...
Eventually, a huge war broke out with fighting in both Europe and the colonies in what is known as the French and Indian War/Seven Years War (1754-1763).
By the end of it, the French gave away New France to the British. They didn't even care really, as they managed to keep some smaller colonies in the Caribbean which produced sugar, which by then made more money than fur. New France was seen as nothing less than "a few acres of snow", to put it in Voltaire's own words. The British now had a lot more territory, but they were faced with the fact that the newly acquired land also came with a whole bunch of settlers... Fearing a possible rebellion if they were too strict, the Brits were lenient and granted many concessions such as religious freedom. Remember that the French are Catholics while the English are Protestants, which was a big deal back then. Either way, they allowed the French settlers to keep their language, their feudal land management system, education system, French civil law code and so on. Many were given the choice of going back to France, but since most of the settlers by then had actually been born there, they decided to stay.
Unrelated but interesting if you're into history: the outcome of that war was one of the catalysts/reasons why the American and French revolutions happened. Both the English and French had to increase their taxes because of the high cost of the war. In the Americas, this created anger among the Americans, who were also angry at the lenient way with which the French had been treated. They assumed that having been defeated, the enemy shouldn't be treated so well. This, combined with the higher taxes prompted the American revolution (1775-1783). The French, just to be dicks towards the English after having lost their colonies to them a few years earlier, helped the Americans a great deal (though Americans like to downplay France's vital role in their success). Helping out the Americans stretched France's finances even further, eventually leading to the French Revolution (1789+). Funny how everything is so interconnected...
Fast forward a few more hundred years to 1900-1950 and you've got a very Francophone province and a very anglophone city of Montreal. Montreal was the biggest city in Canada. All the big businesses were there, big industry and so forth. It's worth noting that even today (I've been there), Montreal has two communities. Anglophone and Francophone (well, lot's of immigrants now as well). The anglophones historically lived on the West side of the island, while the French lived on the East. The English typically owned all the big business and the French, despite being far more populous by then (Catholics have a lot of babies because they don't believe in condoms) worked as laborers and for the most part are were working class. It's worth noting that in those days, most of the signage in Montreal was in English, and all the factory owners/business owners were English. The more numerous French had to speak English with their bosses or in the big shops. Eventually, in the mid to late 1960s (like in many other countries) there was an "awakening" of sorts. The Quebeckers realized that they should be "masters of their own land" and started a big cultural and legislative reform. Laws were passed to make French signage mandatory, immigrants had to learn French instead of English and so on. This wave of nationalism eventually led to two referendums on sovereignty (independence). Both before and after these, many of the big businesses moved their headquarters to Toronto out of fear that they might lose it all if Quebec separated. Many anglophone families (who had lived there for generations) also relocated to other parts of Canada.
However, many families stayed and to this day, there are many anglophones in Montreal and smaller communities in other parts of Quebec. Typically, most people from Montreal range from functionally bilingual to perfectly bilingual. However, you will still find Francophones who either can't speak English, or some Anglophones who can't speak French. This is increasingly rare though (especially if they were born in the city), but you still see it. People like Leonard Cohen and William Shatner are examples of fairly well-known Montreal anglophones. So one can't automatically assume that because someone is from Quebec, they're a Francophone. As I've shown, for historical reasons there are two big communities, at least in the big city. The English community has diminished considerably since 1980, but it's still sizable. To this day, if you're in the east end of Montreal, French is used at home by 95% of people. Downtown, I'd say pretty much everyone is bilingual (shop assistants etc). You can ask for information in either French or English and the people will use whatever language you're comfortable in or whichever you use first when approaching them. In some areas of the western part of the island, 95% of people speak English at home. So while a family may speak French or English at home, most people are bilingual. Population-wise, more people speak French though. Across the city, there are both English and French public schools, colleges and universities. French schools have English as a second language (starting in 3rd grade or so), then Spanish. English schools have French, then Spanish or another language. There are also some totally bilingual schools, where classes are given sometimes in French, sometimes in English and assignments given in the same way. Some of these have been very successful and have waiting lists. People have access to both French and English TV and so on, so it's quite easy for someone born in a 100% English or 100% French family to become fluent in the other language. Outside of Montreal is a whole different story. The rest of the province is 95% French speaking, and many speak pretty bad English, sometimes no English at all.
Canada is officially a bilingual country, so as with Switzerland, laws and federal government papers are available in both languages by law. Technically, Quebec is only French, while New-Brunswick is the only officially bilingual province. However, when I was in Quebec and Ontario, I noticed that both provinces had documents available in both English and French. I watched a Canadian parliamentary talk once and they all speak in either French of English, switching it up, while those who can't speak the other language (usually members of parliament from the west) use ear pieces with instant translations (like the UN). Typically the prime minister must be able to speak French, and most in the past few decades have come from Quebec for that reason. If they can't speak French, they won't get votes from Quebec (which has a high population and thus, vote percentage). The few Canadian PMs who didn't speak French because they came from western provinces had to learn it pretty quickly (Steven Harper). It's expected that the PM speaks both languages fairly fluently.