As non-technical as possible:
If you were to describe anything about who you are and what you were allowed to see or do, each of those things would be something that you were "claiming" to be true, and thus each "thing" on that list would be a "claim".
Anytime you tell someone something about yourself or "claim" that you're allowed to see or do something, you hand them your list of claims. They will verify with an authority that your claims are true and if they are, they'll believe anything on that list of claims. So if you claim that you're Brad Pitt, your list of claims says that you're Brad Pitt, and it was verified with the authority that your claims are all true -- then they will believe that you're Brad Pitt along with anything else in that list.
Claim: what you claim to be true. This can be a piece of information or a description of a permission you're claiming to have. The system to which you present your claims only need to understand what the claim is/means and also be able to verify with the authority.
Authority: The system that puts your list of claims together and signs it which basically says "On my authority, everything in this list is true." So long as the system reading the claims can verify with the authority that the signature is correct, then everything in the list of claims will be considered authentic and true.
Also, let's not call it "claims based authentication", instead let's call it "claims based identity".
Slightly more technical:
So now in this process, you authenticate using some sort of mechanism (username/password, client secret, certificate, etc.) and that gives you a token that proves you are who you say you are. Then, you trade that access token for an ID token. That process will use your identity to find and build a list of claims, sign it and then hand you back an ID token that has all of your claims.
As the authorization step, depending on how it is implemented, the resource will look at your ID token (claims) and then check if you have the necessary claims to access that resource.
So for example, if the resource "CastleBlack/CommandersTower" says that "you have to have access to castle black and be lord commander, then it's going to look at your list of claims to see if both of those things are true.
As you see, "claims" can be anything. It can be a role, it can be a fact, it can be a flag. It's just a list of key-value pairs and the "value" is optional. Sometimes it's just about seeing if the claim exists:
claims : [
{"type": "name", "value": "Jon Snow"},
{"type": "home", "value": "Winterfell, The North, Westeros"},
{"type": "email", "value": "jon@nightswatch-veterans.org"},
{"type": "role", "value": "veteran;deserter;"},
{"type": "department", "value": "none"},
{"type": "allowEntry", "value": "true"},
{"type": "access", "value": "castleblack;eastwatch;"}
]
So if Jon logged in and tries to access the resource described above, he would be denied because, while he is who he says he is and he does have access to castle black, he is no longer lord commander nor does he have explicit access to the commander's tower, and thus cannot implicitly enter the lord commander's tower.
More specifically, "CastleBlack" would probably be a [larger] scope, and each area would be a specific permission, but that's a different discussion.
How each application deals with access is going to be different, but it will use claims to do it.