You know, if it weren't for Braveheart I likely would be one of the majority of non Scottish people that wouldn't even know that this man existed. While a lot of his history is either not known or completely embellished for the sake of making a blockbuster film, he was a pretty damn important dude in Scotland.
While we can't really be certain about the details of the man, nor can we depend on the Hollywood film to be accurate about that, one thing is certain: He was a real thorn in the side of the English and his capture on this day in 1305 was considered a massive victory for England.
In Braveheart, we are led to believe that Wallace was always on the run from the English, but this is not entirely true. There was a period between 1299 and 1304 that the desire to capture and bring Wallace to justice was abandoned by the English. During this time William became an envoy to Europe and spoke on behalf of Scotland, primarily to the leaders of France. At one point, the French united with the Scottish against England but then later the French dropped this alliance because they needed England's help militarily, elsewhere.
The Scottish were on their own at this point and this is around the time that William Wallace was captured in Glasgow and taken to London to stand trial. While the film did depict his public execution as brutal, it was actually much worse than shown in the film. According to historians Wallace was drug behind horses in the streets of London for 5 miles or so before the fun at the torture alter began. Wallace also did not have the chance shown in the film to embrace King Edward and end his suffering with a quick death.
This certainly was not a good way to go.