Vapour Pressure and Equilibrium
The key missing piece is identifying when the system is in equilibrium. When you suddenly increase the volume, the system is temporarily not in equilibrium. In this situation, a few things will happen:


The rate of evaporation won't change for the reason given
The vapor pressure will decrease. As a first approximation, we can use the ideal gas law to see how. $PV=nRT$ can be rearranged as $P=nRT/V$. The number of moles ($n$), gas constant ($R$), and temperature ($T$) will stay the same (this last one in particular isn't really true, but it doesn't change the qualitative result). Then, if the volume increases, the pressure will decrease.
Since the pressure has decreased, the rate at which gaseous molecules bump into the liquid decreases. The ones that do bump into the liquid will condense into the liquid at the same rate. The total condensation rate is the product of those two factors, so the condensation rate will decrease.


Now, since the evaporation rate is the same as it was before, but the condensation rate will go down, we will get a net evaporation. As more liquid becomes gas, the vapor pressure will increase. As the vapor pressure increases, the condensation rate will increase, for the reasons discussed above. Once the vapor pressure is high enough that the condensation rate equals the evaporation rate, then we will have a dynamic equilibrium.