Well it's like how earlier PlayStation 2 models had "backwards compatibility" with PlayStation 1 just because the PS2 had the necessary PS1 hardware to run PS1 games via not an emulator but firmware.
But these "adapter" idea works similarly with a different twist!!! The game actually runs on the "adapter" but its video and audio output used by NES as a bridge between TV to make the output directed to the TV that NES is connected, and then the adapter accepts NES gamepad input to control the game. This is basically how, for example, GBA can run GBC games.
Emulators are needed because of CPU differences beause it's like, consider a game as a Spanish book and CPU as the guy only knows Spanish so naturally the Spanish book can be read by Spanish guy but not an Italian because Italian doesn't understand Spanish in this analogy!!! That's why due to CPU difference emulator needed to act as a real-time translator who translates Spanish book to Italian so Italian guy can understand it. Naturally this process is time and energy consuming and in the process meanings can be lost in translation which makes emulators not the best way to play games on another platform therefore best way is the hardware using the CPU that game is designed for. This is the best way for backwards compatibility!!!!
I mean this is "how these things work" which you can make the "adapter" also have hardware-software emulation but it's overkill lol.