Route53 has nothing to do with HTTPS.
Route53 is just a DNS provider. In shortcut - it maps domain names to IP addresses. There's no encryption on this level of communication.
Encryption comes into place when HTTP request appears.
On one end, there is user with his browser. On second end, there are plenty of options:
- public EC2 instance with HTTP server like Nginx or Apache
- Load Balancer, like AWS Elastic Load Balancer, with certificate and TLS offloading
- API Gateway, with certificate and TLS offloading
As you can see, as long as you use load balancers or API Gateway, you don't need to carry about encryption. If you expose public EC2 with its HTTP server, then you need to install some certificate on it to encrypt the traffic.
Regarding your question about when the encryption takes place - it's all about TLS algorithm. Initially, two parties send messages using their asymmetric public/private keys. The user has its own key pair, so as server/load balancer/whatever. During the initialization, both parties set the symmetric key which will be used to encrypt/decrypt the whole transmission on both sides.