Yes, GAAGO's payment processing system is extremely safe and secure.
Payments are processed by Stripe, which has been audited by a PCI-certified author and is certified to a PCI Level 1 Service Provider. This is the most stringent level of certification available in the payments industry. To accomplish this, they make use of best-in-class security tools and practices to maintain a high level of security.
When entering your card details at the checkout, if you have previously saved your details with Stripe Link you will have the option to retrieve the details from there rather than having to manually type details in again.
For more see Stripe's security information.