Draw closed curves in the space and see which ones are "equivalent"- that is, which ones can be continuously deformed into each other. In the plane, all curves can be deformed into each other so the fundamental group is trivial- it consists of a single element. On a torus, all closed curves are of three kinds- those that can be deformed down to a single point, those that go around the "long" circumference of the torus, and those that go around the "short" circumference of the torus. So the fundamental group of the torus has three members and is isomorphic to the rotation group of a triangle.
In the case of , where l is a line, there are two "equivalence classes" of closed curves- those that go around the line and those that don't.