Wi-Fi assist is as simple as it sounds, when on a Wi-Fi network, your phone will determine if it is capable of loading websites and doing the task you are in the middle of without delay, if it is experiencing a delay, the phone will automatically start combining the Wi-Fi with your cellular data and use both at the same time. Great if you have a large or unlimited amount of data to use and struggle on a poor Wi-Fi at a coffee shop or the gym.
However in other situations, maybe you are in a shared accommodation like a hotel or student accommodation and attempting to FaceTime somebody and your phone decides to start using your 4G over the poor Wi-Fi. This may ramp up data bill significantly if you have not turned Wi-Fi assist off.
When Roaming this is not a problem as Wi-Fi assist will not enable whilst your phone is using roaming abroad, however I would still recommend accessing Wi-Fi wherever possible to avoid any unnecessary data charges.
To enable or disable Wi-Fi assist simply head to settings > mobile data > and scroll right to the bottom to find the toggle for Wi-Fi assist.
NOTE: In the upcoming iOS 9.3 there is a new notification in the Wi-Fi assist toggle that tells you how much data the feature has actually used.