I'll answer this for the case of the torus; hopefully the proof of the general case will be made clear by analogy.
We may view the torus as the quotient of a square, identified opposite edges with the appropriate gluing. Thus the punctured torus can be thought of as a punctured square, which deformation retracts onto its boundary.
For the torus, though, this boundary can be expressed in terms of the two loops that give generate the fundamental group of $\pi_1(T)$. Thus $T$ deformation retracts to the wedge of two circles. This fact can be verified purely geometrically as well.
In general, this method shows how the circles appear (as generators of $\pi_1$ of the surface), and should give you a handle on their number.
If you want another data point, consider the punctured sphere, which obviously deformation retracts to a point (i.e. the wedge of 0 circles).