I read somewhere (https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2014-May/005816.html) that Bitcoin Core is moving to an edge router role. Can anyone tell me why that is? Edge routers in Bitcoin are services like exchanges, online wallets, etc if I'm correct, right?
No. Edge routers are devices (Bitcoin nodes), not services.
Edge router is a term used in networking context: routers that are on the edge (boundary/border) of a larger network. For example a company network that connects to the internet. It often also serves as a firewall to protect that internal network from outside access.
In a similar fashion a Bitcoin core node can function as a firewall of sorts. Outside nodes can only directly communicate to that edge node but not to any other Bitcoin related software inside of the network behind it. Such an edge node would typically not contain a wallet itself, so there is no money directly at stake in case of an attack.
Other nodes on the internal network might contain for example a wallet or they may be running custom software that might contain vulnerabilities or consensus critical bugs that could be exploited if directly accessible to outside attackers.