Think of iOS or Android apps that stream content from various TV services, but dedicated to just that and not all the other functionality of smartphones. ATV is similar in many ways to Roku, Android TV, Amazon Fire. All of these connect to a TV via HDMI and stream content from the internet over WiFi or ethernet connections over your home network.
What sets ATV apart from others is integration with other Apple Devices. Most industry analysts concede, if you have Mac and\or iOS devices, ATV is the logical choice. Why? Because ATV can natively be used to stream music and video content from other Apple devices via Airplay. It shares your iTunes and iCloud account, so purchases, music, video content that are rented or purchased can be played on any of the devices. Photos from your phone will appear in ATV via iCloud photo sync. And if you are putting smart home devices in your home, ATV can be your always on hub for "Home Kit" integration of your smart lights, door locks, window shades, ceiling fans, etc. With Home kit, just invoke Siri on any of your Apple devices and issue a command, from anywhere. Using iCloud, it sends the command to your ATV and executes the command for you.
With some of the services now available on streaming boxes, you can subscribe to a live TV service like cable or satellite, but without the added boxes, remotes, connections, and rental fees. These are typically a set price with no introduction rates, no commitments, cancel anytime unlike cable or satellite that tie you into a 1-2 year commitment.