Firstly, you need to contact your current provider, and ask for a PAC code (Port Authorisation Code). This is valid for 30 days, after which it will expire and you will need to ask for another one.
Then go here:https://giffgaff.com/profile/details
And click where it says "Keep my number" and put the PAC code in.
Your phone will be intermittent on the day of transfer, and can take up to 24 hours to fully move across, so give it time and keep restarting your phone.
You may find some services come across first, and there is a delay for others, like you maybe able to make calls, but not receive them for a time etc.