claim
To say something is true or that it belongs to you.
When you claim something, you state that it's true or that it belongs to you. A scientist might claim that a new medicine works better than older treatments. A student might claim they finished their homework even though their backpack tells a different story. In these cases, claiming means asserting something, whether it's actually true or not.
You can also claim something that's rightfully yours. After winning a contest, you'd claim your prize by showing up to collect it. When travelers arrive at the airport, they claim their luggage from the baggage carousel. Native American tribes have fought for decades to claim lands that were taken from them unjustly. Miners during the Gold Rush would stake a claim to a particular area, marking it as theirs to dig.
A claim (the noun) is the thing being stated or requested. An insurance claim asks the company to pay for damage or loss. A false claim is a statement that isn't true. In arguments or essays, you make claims and then support them with evidence.
The key thing about claims is that they need backing up. Anyone can claim anything, but whether others believe you depends on your evidence and trustworthiness. Extraordinary claims require extraordinary evidence.