A chargeback happens when a customer goes to their credit card company and requests that a charge be reversed - that is, they want their money back. This can happen for legitimate reasons if a customer didn't make a purchase themselves for some reason (perhaps their credit card information was stolen and someone else made the purchase).
It can also happen when a customer believes they deserve a refund but the merchant (i.e. you) is not giving them one. This can happen because the merchant is not responding to their request for a refund (this is why it's important to make it easy for your customers to contact you) or perhaps the merchant doesn't believe the customer deserves a refund.
However, anyone can request a chargeback on anything, and some people take advantage of this. This is known as "chargeback fraud", which you can read more about here on Wikipedia.
How to avoid chargebacks in the first place
In short, make it easy to get in touch with you.
Are you monitoring the email address that you're using to send confirmation emails from Brandibble? Customers often reply to their confirmation emails when they have issues or would like refunds for their orders. If needed, you can change the sender email addresses and the content of the email confirmations by going to Settings > Email Confirmations in Brandibble.
Aside from that, having a contact form or contact information clearly listed on your website is also important.
What to do about it when it happens
When you receive a chargeback from your credit card processor (CardConnect or WorldPay / Vantiv), the first thing you should do is contact the customer in order to figure out why they requested the chargeback. It's quite possible that instead of going to you and requesting a refund, they went directly to their credit card provider and claimed the charge on their card was fraudulent. Customers might do this if they are unable to get in touch with a restaurant to request a refund, and credit card companies will honor this request (giving the customer the benefit of the doubt).
So if you reach out to the customer to figure out why they requested the refund, you can often resolve these issues. But if you're unable to reach the customer, you can dispute the chargeback and get your money back.
In order to do this, you'll need to prove to your payment processor that the order was delivered to and accepted by the customer, and that they made no attempt to contact you directly to request a refund (or if they did, why you refused to grant the refund).
You should reach out to CardConnect and WorldPay directly to figure out exactly what they need in terms of evidence. The Brandibble email confirmation or printable order summary is a good place to start - this shows it was a legitimate purchase made by the customer (with their name on it). Any proof of delivery / acceptance of the order would also likely be helpful, but you should deal with CardConnect and WorldPay directly on these matters to see what else they need.