What does "BreakJoints()" do other than the obvious?

All Snaps and Weld and Glue and Hinges &c are parts inside the JointsService.

The JointsService is just the place where all snaps are supposed to go.

When you call BreakJoints on a part or model, the game deletes all of the snaps involving those parts (most of which are in the invisible JointsService, but act just like other Welds or whatever that you've created)

The surfaces of parts themselves don't actually mean much: they only decide how MakeJoints() picks which types of snaps and which parts to snap together.

EDIT:

To find joints on a part is a little difficult. While the canonical place to put JointInstances is the JointsService, lots of scripts that add joints put them elsewhere.

To accommodate that, you would have to recursively scan the Workspace as well. To just find the joints that are made by MakeJoints() (from ROBLOX handling touching surfaces that snap) we just search through the JointsService:

function findJoints(part)
	local j = {};
	for _, joint in pairs(game.JointsService:GetChildren()) do
		if joint:IsA("JointInstance") then
			if joint.Part0 == part or joint.Part1 == part then
				table.insert(j, joint);
			end
		end
	end
	return j;
end