ELI5 version: Chik-fil-A corporation "donates" a significant portion of its corporate revenue to a shady clearinghouse calling itself a charity.
That pseudo-charity was FOUNDED by Truett Cathy, the owner of the restaurant corp. It's his baby. It's an aggregator for donations from many sources - churches, corporations, individuals, etc. It then parcels out that cash to organizations that further its religious agenda.
Religious foster homes, religious scholarships, religious "missions" to foreign countries, and religious anti-gay hate groups -- all of these have received money from Truett Cathy's foundation.
The problem: Dan, Truett's son and CEO of Chik-Fil-A got on the radio and said how proud he was to support anti-equality legislation and political groups. Predictably, people got a bit upset. Presumably, gay people who ate at Chik-Fil-A felt a bit.. disregarded. Cathy initially stuck to his Bible, then started to backtrack, saying he "vauled" all his customers.
Problem number two: He didn't stop funding anti-gay political action, and it took a LOT of protests and media mockery before that promise was extorted from him.
Problem number THREE: When people went digging, as they do after a major foot-in-mouth scandal, they found out something a lot more sinister than just political flyers: Cathy's Foundation had been funding honest-to-God hate groups, of the sort that preach genetic inferiority of gays, advocate forcibly torturing the gay out of teenage kids in discredited internment camps, and have traveled overseas to incite hatred, resulting in violence, against LGBT communities in Belize, Uganda, and Brazil.
How did Chik-Fil-A react to the public outcry? They fairly successfully turned public debate away from their material support of violent hate crimes, at home and abroad, by re-framing the problem as a matter of religious opinion and free speech. Now everyone's forgotten about the torture inflicted by Exodus Internation, on WinShape's dime. Everyone's saying what you said -- it's about his comments. And his family values charities.
Sorry everyone, but the only family values that this shit represents might be the Manson Family.
TL;DR: Chik-Fil-A may have indirectly funded religious terrorism overseas, by giving funds that enabled hate-missions in Uganda and elsewhere, resulting in violence and political destabilization. They definitely funded child abuse of gay youth by contributing operating funds to "pray away the gay" internment camps. This isn't a "speech" issue, it's a question of material support for overt criminal acts.