Typically, angular momentum is just linear momentum plus an attachment. Imagine two weights spinning around each other in a circle, tied together with a rope. At a moment in time, each weight has some linear momentum, each in the opposite direction. The system has angular momentum because part of it has linear momentum in one direction and another part has linear momentum in another direction. In classical mechanics, it's all like that, with different pieces in different locations moving in different directions. But with a particle, there's only one piece. If it has angular momentum, it's gotta be a property of that piece, instead of the more emergent property of a system made of multiple pieces. Hence "intrinsic."