It looks to me like the guys in suits shoved all of the protesters off the stage. I can't tell from this brief clip if those are regular attendees or something else but the protesters seem to be getting the worst of it.

Why are they targeting Sen. Murkowski? The rest of the group's short thread on X explains.

Murkowski played a lead role in persuading Biden to approve the Willow Project. This monstrosity will transport 200,000 barrels of oil every single day. This is hideous. This is cruel and barbaric beyond description. (3/5)

— Climate Defiance (@ClimateDefiance) April 18, 2024

Murkowski claims to support climate legislation, but refused to step in and pass Build Back Better as Joe Manchin held the bill hostage for months. How does this woman sleep at night? How does she live with herself? Is there no limit to her depravity? Is there no limit?
(5/5)

— Climate Defiance (@ClimateDefiance) April 18, 2024

As for that last complaint, Build Back Better was held up by Joe Manchin because he was concerned about over-spending at a time when inflation was rising. Democrats wanted to spend $3 billion and Manchin wanted to spend about half of that amount. And it turns out he was right to be concerned. Steve Rattner, no right-wing conservative, later said that inflation would have been even worse if not for Joe Manchin blocking BBB.

But Climate Defiance doesn't care about anything like that, much less about disrupting an event. Their website makes clear they are all about direct action:

We have blockaded the White House Correspondents Dinner. Stopped play at the Congressional Softball Game. Crashed the Fed Symposium in Jackson Hole, the world’s largest gathering of central bankers. We have confronted too many Senators and Cabinet Secretaries to count.  

We will continue to engage in disruptive, nonviolent direct action until our political leaders, up to the President, meet this moment with the urgency it deserves.

Here's their list of demands/goals:

End fossil fuel extraction on federal lands and waters.

End sacrifice zones and usher in a just transition for those most impacted.

Stand in solidarity with our sister-struggles for racial and economic justice...

Move the Overton Window (the accepted terms of debate).

Make clear to Democratic lawmakers that the youth vote will only deliver for them if they deliver for us.

They were recently featured in an article at Rolling Stone:

“In one year, we’ve reached 70 million people on Twitter, with $0 in paid ads,” said [Executive Director Michael] Greenberg, noting his organization’s activism has been covered in The New York Times, Bloomberg, The Guardian, and by this reporter in Rolling Stone. “This is absolutely having an impact,” he said. “And if we can do this in one year, on pretty much just the East Coast, imagine what we could do if we had twice the budget. We could be doing this in Chicago, L.A., and Houston — in the belly of the beast of the oil industry. Imagine what we could do if we had a real vibrant, distributed organizing program, so that people in small cities and [in] Omaha, Denver, and Minneapolis have the support they need to start their own chapters.”

Greenberg said that in just one year, Climate Defiance hounded several prominent officials so badly they quit their jobs. He noted the group protested Interior Department official Tommy Beaudreau, or “the climate criminal who personally signed off on the Willow Project, [and] 15 days later he resigned.” 

The group claims this is working but I wonder how well it is working on Sen. Murkowski. She has recently made it clear that she's not averse to leaving the GOP behind. Many Republicans would say good riddance but there's also a real chance her departure could hurt the party if it has a narrow advantage in the Senate next year. New York magazine recently suggested she could wind up in control the balance of power.

If as the oddsmakers currently expect, Republicans flip control of the Senate by a couple of seats, Murkowski could stay exactly where she is as a senior appropriator and member of the Senate Energy and Natural Resources Committee, important to energy-exporting Alaska (she’s also the ranking Republican on the anachronistically named Senate Indian Affairs Committee, another panel especially significant to Alaskans). But if it’s a 50-50 Senate or if Republicans need her vote to take control, she could easily declare herself an independent and offer up her support to the party that offers her or her state the most goodies. She’s not up for reelection until 2028, when she will be 70, not that old by current standards. And while Alaska Republicans are trying to get voters to overturn the Top Four ranked-choice vote system that they approved in 2020, so long as it exists, Murkowski is a very solid bet for perpetual reelection. She is well positioned to become queen of the Senate any November now.

If that's how things are trending, the last thing Democrats want right now is some left-wing activist group reminding Murkowski that there are extremists on the left too and they can be very unpleasant. If she thinks becoming an independent would spare her from Climate Defiance and their ilk, she's wrong. If anything, it may cause them to double down on their direct action toward her. The far left usually targets those closest to them ideologically rather than those farther away.