I got the free app for Apple TV on my phone. Works great. But I can't get it to stream on my television. Do I need to get n Apple TV device to do that?

That depends.

• Some modern TVs are AirPlay-enabled. Then you can stream from your iPhone to the TV.
• Some modern TVs can have the Apple TV app among their smart apps. Then you can watch without using your phone.
• Other TVs will need to have a streaming device connected, such as the Apple TV box, or a supported third party streaming device.

See all AirPlay 2-enabled smart TVs - Apple

All the ways to watch Apple TV+ and other content in the Apple TV app - Apple Support

Apple TV app-supported smart TVs and third party streaming devices - Apple