If you are using the Brainscape app on your phone or tablet, you will need a connection to download the app, sign in, and download any flashcards you'd like to study. The amount of data used depends on how "heavy" your flashcards are. (e.g. If they have a lot of images or audio, they will use up more data.)
You can always decrease Brainscape's impact on your carrier's data plan by simply connecting to a WiFi source.
That said, once you have downloaded the flashcards you need, you can totally use Brainscape without a connection (unless you need to create or edit flashcards, which need to be synced with the server for backup). All of your study data will "live" on your device, until you reconnect with a wireless internet data source.
See also: Does Brainscape drain my phone's battery? (<-- No, it doesn't.)