This is all by r/aliment on r/soccer. I did none of the research, but I didn't know any of this and I thought it was a great read.

Who did it and why?
In short, extremist football club supporters ('hooligans', 'ultras') who are unhappy with how the Croatian Football Federation (FF) is run and the (supposed) corruption within. They don't like the leadership with whom they are at war for years now. What happened today is their attempt at trying to force their way, embarrass the Croatian FF officials and incite outrage in the football world. By doing so, they hoped to disqualify the Croatian national football team from the EURO competition. They think this (amongst other things) will put enough pressure on the people they want to step down to step down.
This is not the first time they've done something like this and they are adamant to continue until their requests are satisfied. You might remember Croatia being punished/fined/disqualified 15+ times in the recent years due to disturbances such as this one (riots, clashes with the police, racism, pitch invasions, etc). Especially prominent was the swastika mowed into the Split stadium field the night before the EURO 2016 qualifier match against Italy.
Again, it is important to understand that these are not random acts of boredom or drunkenness; they are planned and precisely orchestrated to further an agenda.

Who specifically in the Croatian FF is the problem for them?
Mainly, Zdravko Mamic and Davor Suker.
Zdravko is the de facto director of the Croatian FF and Dinamo Zagreb football club.
Davor is the formal president of the Croatian FF and a member of the UEFA Executive Committee.
Shortly, the two most powerful characters in the world of Croatian football.
Zdravko is being accused of setting up unfair player contracts, purchasing referee decisions, tax evasion, ignoring supporters' wishes to participate in the club decision making, and interfering with the Croatian national football team. Some media outlets have published this picture of Zdravko giving instructions to the coach Ante Cacic before today's game. Some people claim he is the decision maker of who gets to be the coach of the national team and which players get to play. He is also criticised of selling hundreds of millions of euros worth of football potential from 'his' club (Modric, Corluka, Mandzukic, Eduardo da Silva, Sammir, Halilovic, Kovacic, Pranjic, Badelj, etc.) instead of increasing squad strength by keeping the players and competing in the major European tournaments (he defends himself by saying he creates great players and that he props up the Croatian economy by paying millions in taxes as well). As a result the club is seen as the transition point between Croatian meagreness and European superstar clubs; the general sentiment being that if you are a player and you want to succeed, you have to go through Mamic.
Davor is similarly accused of corruption and being a right hand to/of Zdravko.

The crazy thing that I didn't know, is that Mamic allegedly forces Zagreb players who sign with the team, to sign some kind of personal contract that forces them to pay 20-25 percent of their wages throughout their ENTIRE career to him. And if they refuse, then they don't make the national team and Croatian players families' safety is at risk. One of these such players is none other than Luka Modric and an example of one who refused to sign the 'private' contract is Dejan Lovren.

TL/DR Frick Croatia's FA.

This post was edited on 6/18/16 at 6:12 am