The CD is not the game. It's not like a console. The game is on the CD, though. Some PC games require the CD to be in the drive when you play (even though it's not running from the CD at all, it's just to make sure you own it), however, with the rise of digital distribution (Steam, GoG, etc) that's no longer necessary and very few games still keep the practice. Arena and Daggerfall do not require the CD to be in the drive to play it after you've installed it. Morrowind and Oblivion, on the other hand, do. But if you bought the Anthology you also got Steam keys for Morrowind and Oblivion so you don't need the CD to be in the drive for those either in that case. Arena and Daggerfall are not on Steam, though.
TL;DR, no you don't need the CD in the drive when you play Arena or Daggerfall (as long as you've installed it).