There are two main reasons explaining that you can't use Internet Services on your BlackBerry:
You did not subscribe to the BlackBerry add-on
When you are using a BlackBerry on giffgaff, you can make calls and send texts whatever the credit or goodybag you choose. However, to use BlackBerry Services, such as BlackBerry Messenger, you will need to buy a BlackBerry Plan goodybag which includes the BlackBerry addon (unless your device is running BlackBerry 10 in which case you do not need a BlackBerry Plan).
You will need the BlackBerry Plan with the inclusive Blackberry add-on to use the following services:
You can find more information on how to buy the BlackBerry Plan and add-on here.
Your phone settings are incorrect
If you have bought a BlackBerry add-on but are still unable to use any of the BlackBerry Services - or use Internet - the issue is probably coming from your settings.
Check this article to see the different methods you can use to update your phone settings. If you phone settigns are correct, make sure that your BlackBerry ID is well recognised on your device.