Following Mikasa's answer, I also agree that this is a broad question with a variety of reasons and is also quite opinionated. Some people do just protest because, well, they can. However, as a British person, I'll base this answer on the EU Referendum and why the protests for the outcome were particularly large.
So, first, let's dive in to our political spectrum - the range of opinion in our referendum:
(I've attempted to use neutral colouring here, but for clarity, I voted remain. See the following image for my position).
Using the above information as well as other generally available results, I'll make the following points:
- The Referendum is "non-binding", meaning the result doesn't automatically become law.
- It's a very tight referendum; the majority for leave was pretty small.
- Large chunks of people have relatively central views.
- It felt like a dark day for Europe, so people wanted to essentially say to Europe that "we're still your friends".
- The views of the winners were quite extreme. This is particularly apparent when we take a look at where major political players stand:
From my point of view, they're a really long way away. The kinds of policies that some people in the new Cabinet are suggesting are deeply concerning to me. We've got a new Cabinet that nobody voted for - it wasn't an election. These are the kinds of things that a protester is thinking about.
So, now consider the amount of people that are over to the right of the Governments position on our charts. That's a lot of people who's opinion is mostly being ignored by these "hard Brexit" suggestions. In general the further away you are, the angrier/ more disappointed you get.
This one's very much an EU Referendum only thing. Notice how Parliament - our elected representitives - are on the other side from the current Cabinet (the currently unelected executive branch). As a result, the Cabinet suggested that the EU bills would not pass through Parliament at all, on the assumption that Parliament would simply block it. People aren't happy that Parliament is essentially being ignored. This has lead to a partial Constitutional Crisis. It's a particularly interesting situation but further discussion on this note is outside the scope of this answer.
Next we have a more subtle cause - location. London is a major trader with Mainland Europe, particularly with financial services, and so naturally, it strongly voted to remain. London is also where Parliament and the largest protests were located - in effect, the strongest remain voters didn't have far to travel to make their opinion at least publicly visible.
Boris Johnson, displayed on the chart above, is also the previous Mayor of London. He is of course well aware of The City's (London) ties to Europe. His seemingly abrupt change from remain to leave felt like he had essentially stabbed The City - his city - in the back, in favour of his own political gains, as he was presumed to succeed David Cameron as the next Prime Minister. As a result, many protests occurred right outside his house.
Of course in any protest it helps to read the signs to see what people's concerns are. The common ones in this case (and in the US) are that people feel like they're losing freedoms and are gaining insecurity. As a common quote goes, "freedom must be fought for". These kinds of fundamental feelings will also push people to express opposition.
There's a lot of factors at play here - including many that I haven't included - but it can be summarised as a tight election that resulted in a winner with relatively extreme views, seemingly leaving many people out. The general feel of the protests is a strange mixture of openness (we're still your friends!) and the loss of it.
Given this, I would say they're actually being pro-democracy as they're trying to pull the winning side towards the central ground where the overall opinion actually sits.