What reason could cause AWS RDS connections to spike
If you run show processlist; it'll show you all the connections that are running against your DB.
Running show status like 'Conn%' can show how many connections are active

You may think that a high number of connections would increase the CPU but ultimately it depends what those connections are trying to accomplish, if they aren't doing much but are preventing you're main application from running their connections then the normal processing level will go down while you are not running your normal processes

The annoying thing is to be able to run the show processlist command you need to actually be connected to the DB which you wont be able to as you're at max connections.

A slight work around is to increase your max connections (if you're able to) and set up some monitoring somewhere that runs every so often (every 5 minutes would have caught that) and when you're over 50 connections, run a command that dumps out all the active connections somewhere so you can review later.