Someone over here, stateside, might be excused for not keeping up with the politics in Europe. However, if you are a fan of the USMNT, you might want to take a look because the Brexit may end up hurting American players.
So a quick review of what the Brexit actually is. Brexit is the (awful-sounding) short-hand for the British exit from the European Union. The country held a referendum to decide whether the voters wanted to stay or leave, and they voted to leave. There are a number of different reasons people have for both staying and leaving. I'm not going to go through the actual political issues, except the ones that clearly will affect Americans.
The biggest issue for American soccer players (and other expats) is the changes in visa laws. In order to work in Britain as a foreign national, you need a visa issued by the British government that says you can work in the country. The European Union has a policy where any citizen of any member nation has the ability to move freely between other member nations. That has meant that players like Danny Williams and Gedion Zelalem have been able to go to England through German passports.
If you don't have a passport from a Euro nation, you have to go through the Football Association's process, which can be a bit arbitrary. Players are required to represent nations that have played 3/4 of all competitive matches in the last two years, and the nation must be ranked in the top 70. If a player does not meet those requirements, they can appeal and the FA will decide if they are a compelling enough talent. You can take a look at the rules here. A number of American players have been denied in the past. The list include DeMarcus Beasley, Brad Friedel, Cobi Jones, Brad Guzan, and, more recently, Juan Agudelo.
A European passport used to mean that a player could make the jump into England without the complications. Players like Zelalem carry European passports (usually through their parents or grandparents, though, in Zelelam's case, through his birth nation), allowing them to be scouted and considered by Premier League teams, in this case, Arsenal. Removing that freedom of migration makes the process more difficult. And this doesn't apply just to the Premier League. This hurts players trying to move to any of the four nations in the United Kingdom, including Scotland.
There will be other effects on soccer and Britain at large. When votes started to come in that the Leave movement had won, the British currency, the pound, experienced its biggest drop in decades. The results of the referendum technically are not binding. The British Parliament has responded by calling for elections in October, so the voters for the Leave movement need to also vote in representatives that will act accordingly. A lot has happened in the past 24 hours, but this issue won't be resolved any time soon.