Do I need Apple TV to stream wirelessly to my TV?
There are two options which will suit what you want


Any AirPlay video receiver will let you stream wirelessly. (You can stream the audio and video, or just the video and use headphones) AppleTV is one such receiver and by virtue of it being on the market the longest, it has the most users with experience as to how it works.
Buy a longer cable and link up physically.


On a personal note, I'd be in favour of the longer cable option, as it offers slightly more compatibility (a friends laptop, perhaps) and costs way less. That said, Apple TV's are awesome. I'd also recommend using either HDMI or Thunderbolt/displayport.

In short, yes, an Apple TV will do what you want.