Original: An open string looks like a short line, so its worldsheet will look like a strip or, more generally, a Riemann surface with a boundary.
Simplified: A closed string looks like a small loop, so its worldsheet will look like a pipe or, more generally, a Riemann surface (a two-dimensional oriented manifold) with no boundaries (i.e. no edge). An open string doing the same thing will have its worldsheet looking like a ring connected to two strips.