In the US, patriotic tradition holds that you should stand (and remove your hat) during the playing of the national anthem.
On one level, it really is that simple, but to fully understand everything happening now we need some further context:
During the 2016 NFL preseason, San Francisco quarterback Colin Kaepernick remained seated on the bench during the national anthem. He did this quietly for several games without attracting attention, but when it was noticed, he explained why he did it:
I am not going to stand up to show pride in a flag for a country that oppresses black people and people of color. To me, this is bigger than football and it would be selfish on my part to look the other way. There are bodies in the street and people getting paid leave and getting away with murder.
Kaepernick's protest, then, was in solidarity with the Black Lives Matter movement in the US, which was started in response to the high profile deaths of (primarily) men of color at the hands of police.
As you can imagine, his actions and his statement garnered a lot of controversy, and in the intervening year other players (and even athletes in other sports) have joined in, sitting or kneeling during the anthem.
At the time, then, those that supported the Black Lives Matter movement (generally) supported athletes sitting or kneeling during the anthem as a symbol of protest, and those that opposed the Black Lives Matter movement (generally) opposed it.
This wasn't universally true, but I think it is an accurate generalization.
Donald Trump, both during his campaign and his presidency, has vocally opposed the Black Lives Matter movement:
Trump has also exchanged jabs with Kaepernick specifically:
In the midst of explaining his actions to reporters, Kaepernick called then-candidate Trump "openly racist."
Trump wasted little time in firing back when asked during a radio interview about Kaepernick's protest.
"I think it's personally not a good thing. I think it's a terrible thing," Trump said at the time. "And, you know, maybe he should find a country that works better for him. Let him try. It won't happen."
This past Friday, though, Trump went even further at a rally in Alabama:
Wouldn't you love to see one of these NFL owners, when somebody disrespects our flag, to say, "Get that son of a bitch off the field right now, out. He's fired. He's fired!"
Now he's certainly not the first to express this kind of opinion, but it is a little different coming from the "bully pulpit" of the President of the United States.
Trump's statement (and other similar tweets about the NBA) received widespread attention and criticism from many groups. Many players, owners, and teams, even those that hadn't previously taken a position, spoke out in support of their player's rights to express their views. Free speech groups objected to the President appearing to condemn a constitutionally-protected protest. Entire NFL teams knelt or joined arms, or stayed in the locker room during the anthem to signify their disagreement with what the president said.
Note that these actions this weekend were largely about the President's statement, and don't necessarily mean that all those players and owners and coaches are in agreement with Black Lives Matter.
Many veterans, for example, publicly stated that they supported the right to protest, even if they didn't agree with what was being protested.
So there is a lot of talk about free speech, and player contracts, and "honoring veterans", and all sorts of things, but the original reason (police violence and racial justice in the US) is still a major factor in whether people support or object to these protests.
EDIT: Colin Kaepernick's SF teammate, Eric Reid, has written an editorial explaining why they chose to kneel:
I approached Colin the Saturday before our next game to discuss how I could get involved with the cause but also how we could make a more powerful and positive impact on the social justice movement. We spoke at length about many of the issues that face our community, including systemic oppression against people of color, police brutality and the criminal justice system. We also discussed how we could use our platform, provided to us by being professional athletes in the N.F.L., to speak for those who are voiceless.
After hours of careful consideration, and even a visit from Nate Boyer, a retired Green Beret and former N.F.L. player, we came to the conclusion that we should kneel, rather than sit, the next day during the anthem as a peaceful protest. We chose to kneel because it’s a respectful gesture. I remember thinking our posture was like a flag flown at half-mast to mark a tragedy.
It baffles me that our protest is still being misconstrued as disrespectful to the country, flag and military personnel. We chose it because it’s exactly the opposite. It has always been my understanding that the brave men and women who fought and died for our country did so to ensure that we could live in a fair and free society, which includes the right to speak out in protest.