WiFi Assist is supposed to switch to Cellular when your WiFi signal is weak
The problem a lot of people are getting into is that the feature can overcompensate a little bit too strongly and it results in a LOT of data usage.
If you're not in an unlimited data plan, you might wanna check your data usage and turn it off if it's eating through your allotment.
Here's a quick video showing you how to turn off that feature
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Source: http://www.engadget.com/2015/09/28/ios-9-wifi-assist/