Airplane mode turns off all your phone's radio connections (cellular, Bluetooth, and Wi-Fi), but apps that don't require an internet connection will continue to use data. Your best bet to avoid expensive roaming charges is to turn off data roaming or — even better — use a prepaid data plan.
Check your Cellular/Mobile Data options and turn off Data Roaming. In the Settings app, tap Cellular, or Cellular Data, or Mobile Data. Turn on Cellular/Mobile Data, then tap Cellular/Mobile Data Options. Turn off Data Roaming.