Wi-Fi Assist kicks in when your Wi-Fi signal is low or weak. Instead of just choking when trying to load a webpage, for example, Wi-Fi Assist will switch to using the cellular data connection to complete the download. Yes, this might result in increased data usage. For my personal use, I've seen Wi-Fi Assist being used only rarely and I can't really see enough of an increase in data usage that will cost me more money. I always stay well within my data allowance.
You can determine is Wi-Fi Assist is being used by checking the status bar on your iPhone. If you are connected to a Wi-Fi network and the Wi-Fi fan disappears and is replaced by LTE, 3G etc. you are using Wi-Fi Assist.
You can turn Wi-Fi Assist off in the Settings app.
Settings > Cellular > Scroll all the way to the bottom to find > Wi-Fi Assist < Turn it off if desired.