I don't get why it's so important for many brits like us to get so bent out of shape about another countries choice of leader. It's not like we had any choice or can do anything about it.
It's less about the leader and more about what he is doing (or being told to do..) and the protests... well that's many fold frankly.
Firstly, where the US goes the UK tends to follow - given that Brexit means we need to secure trade deals we are even more likely to suck up to the USA; this is already in evidence given how long it took May to say something about the ban (any other country, or other time, it wouldn't require pulling teeth) and how quickly she offered a state visit. (Obama was in his second term, Bush and Clinton didn't get one; Cheeto Hitler - 7 fucking days!). So the protests here are as much against Cheeto Hitler as they are directed at our own government in a "Don't try this shit here" way.
Secondly, it is about support. It is supporting those in the states who are protesting so they don't lay down and take it either. To know that 'hey, many of us agree his actions are bullshit and we stand with you'. Given the people in power, a mixture of White Supremacists and Religious wackjobs, I suspect those protesting like the support and knowing that others stand with them to protest this dark direction the USA is heading in. (I wondered to myself earlier, if the world was as connected as it is now back when Hitler was rising to power if the protests might have stopped him, when people in the country knew others stood with them and that it wasn't them alone so they had to go along with it.)
However it isn't just about giving American's support as their Government starts to go to shit; most people realise that the ban which triggered all this isn't a sane move (heck, the US intelligence agencies didn't ask for it!), it is the perfect propaganda tool for people like ISIS to point at and say "See! The West hates you!" and with everything directed at Muslims these days for many it could be the straw which breaks the camel's back. The protests show that not everyone thinks that and that message also gets out to places like Iraq.
Indeed, I saw something earlier where someone serving in Iraq was talking to someone they were working with and they said (paraphrased slightly as I don't have a direct quote to hand) "We thought that having worked along side you, thought along side you, and tried to make the world safer along side you that America would be more welcoming to us..." - a pause, and the American in question thought they knew what was coming next - "... and we see that it is from the protests.".
The message gets out there, sitting idly by and making no noise just enforces the notion that the West doesn't care and thinks they are all terrorists.
Which leads in to my next point; geo-political stability.
For the longest time the world was getting better in many ways; China, while not perfect, were in the fold (tense in recent times, but still talking). Russia was doing it's own batshit thing but at least the USA could counter them. Iraq was proving a focal point for anti-ISIS actions with the likes of the USA and UK doing something to improve the clusterfuck they created working along side the locals. Even Iran was coming back in to the fold and becoming less of a crazy state.
Things weren't perfect of course, battle grounds persist and dickmoves continue to happen but there was a degree of healing, however slowly.
We are now 12 days in to Puppet Hitler's term - he has pissed off the Chinese, he has pissed off Iraq, he has pissed off Iran. He is no longer a good balance against Russia. And everything is sliding backwards.
So you protest, you protest in your own backyard to stop yours becoming as bad as your neighbours and stop the whole place going to shit - because ultimately the shit the USA is flinging will impact us all; there is an old adage - if the USA sneezes we all catch a cold. It is as true then as it was when I first heard it in the 80s.
But ultimately the reason to protest is because we are all human. Countries and boarders, it's all PR and propaganda - you protest and make a noise because fellow humans are suffering. The moment you lose sight of that, the moment you think 'the suffering of another person isn't my problem because they aren't born here', that's the moment you fail any future humanity might have in this universe because, if there is one thing I'm sure about, we don't stand a chance as a species if we don't pull the fuck together...
... and right now, with how the world is going, I don't see that happening before we have another global war; maybe after that those who are left will learn to work together.
Ya know, assuming anyone is left...