A router is what its name implies... It routes network and internet traffic.
You need an internet source if you want connection to the outside world. An internet service provider (ISP) is provided typically by cable and phone companies. A modem is a device that is necessary to host your internet connection to your home or business. It's the ISP's terminus to your location. Some modems can also double as a router, and will have several ethernet ports on it to handle more networking device connections. A simple modem (with no router capability) typically will only have two ports. One port for the incoming internet connection from the outside, and one port to connect your internal networking devices like a router or switch.
The router can handle network traffic among internally connected devices, and it can do this to have devices talking to each other, whether you're connected to the internet or not. A router can handle both wired and wireless connected devices, as long as the router clearly is labeled as a wireless or wifi router. The ASUS RT-AC5300 can manage both wired and wireless device traffic.
It's my guess that you want your internal network devices, not only connected and interacting with each other, but you probably also want connectivity to the internet.
The ASUS RT-AC5300 is not a modem. The ASUS should receive it's internet connection from a modem to give the rest of your network access to the internet.
If you have internet connectivity at your location now, it's most likely being served by a modem provided by your ISP, or by a modem you bought yourself.