 Solution

Choose an essential simple closed curve $c$ in the oriented topological surface and an annular neighborhood $N(c)\cong S^1\times [0,1]$. A positive Dehn twist about $c$ is represented on this annulus bywhere $\rho$ is zero near $0$, one near $1$, and increases between them. Since a full turn is the identity on the boundary circle, this map is the identity near both boundary components of the annulus and extends by the identity outside it. It preserves orientation and turns one end relative to the other by one full turn. An inverse twist uses $-\rho$.

More intrinsically, $h$ is a Dehn twist if its mapping class is that of this annulus-supported homeomorphism, for some curve $c$ and the chosen handedness. Choices of annular coordinates and interpolation give isotopic maps. The genus hypothesis permits essential curves, but the definition also makes sense for boundary-parallel curves on surfaces with boundary.