Why do I have to install CLT when Xcode is installed? On a fresh Mavericks, I installed Xcode via the App store (accepted license agreement and enabled developer mode). After that, Command Line Tools (CLT) are found (checked via xcode-select -p). Installing Homebrew as explained on their webpage (ruby -e "$(curl -fsSL https://raw.github.com/Homebrew/homebrew/go/install)") will result in the request to install the CLT. Why do I have to install them? I thought they are installed together with Xcode.
You have to install them because, starting with version 4.3, Xcode is released as a sandboxed, single app bundle.

Before that, Xcode was distributed as an installer called Install Xcode.app. The installer had access to the whole filesystem and created a /Developer directory, where frameworks and additional tools where placed, and installed the command line tools in /usr/bin.

This is no longer the case. Apple chose to make XCode a standard Mac App Store app that installs in /Applications and requires user intervention to install any additional components outside the bundle container.