Units and some equations
The Hubble constant effectively has a unit of inverse time only: ${\rm s}^{-1}$:
It is usually given as ${\rm km}\cdot{\rm s}^{-1}{\rm Mpc}^{-1}$. Yet looking at it, km and Mpc are both units of length, so they cancel eachother out when converted to the same unit. Thus add the appropriate unit conversion from km to Mpc and it remains an inverse time only:
$$t = \frac{1}{H_0} = \frac{\rm Mpc}{60 {\rm km}\cdot{\rm s}^{-1}} = \frac{3\cdot 10^{22} {\rm m}{\rm s}}{60\cdot 1000 {\rm m}} = 5\cdot 10^{17}{\rm s} = 15{\rm Ga}$$
Add dimensionless factors to the equation to account for different cosmological models and inflation theories. This is anyway only a ballpark number - but an easy one to get for the age of the universe.