What happens when AWS RDS Proxy can't get a connection?

What happens specifically when the AWS RDS Database that's being accessed through an RDS Proxy doesn't have any connections to give? I've found the following information in the RDS Proxy user guide:

RDS Proxy queues or throttles application connections that can't be served immediately from the pool of connections. Although latencies might increase, your application can continue to scale without abruptly failing or overwhelming the database. If connection requests exceed the limits you specify, RDS Proxy rejects application connections (that is, it sheds load). At the same time, it maintains predictable performance for the load that can be served with the available capacity.

Every article, blog post, stack overflow post or otherwise I've read on the subject just parrots this without providing any specifics on what this behaviour is. What exactly happens when 'RDS Proxy rejects application connections'? When there are no DB connections left to give, does the request just hang until a connection becomes available or a timeout is hit? What does the response from the Proxy look like in this case? Is there a list somewhere of these values?

For context, in my code there's a synchronous call to an API Gateway endpoint which invokes a Lambda that then connected to an AWS RDS Database instance.

Thanks!