First, repo init creates the .repo directory, clones the git repository https://android.googlesource.com/tools/repo to .repo/repo, and the git repository specified with the -u option to a bare repository at .repo/manifests.git.  After that, it creates the .repo/manifests directory, converts it into a git repository through creating symbolic links from .repo/manifests/.git to .repo/manifests.git.  It then checks out the branch specified in -b, and creates a symbolic link .repo/manifest.xml pointing to the specified file (-m option) in .repo/manifests, by default .repo/manifests/default.xml.  
Roughly as follows:
  repo init -u $URL -b $BRANCH -m $MANIFEST
  --------------------
  mkdir .repo; cd .repo
  git clone https://android.googlesource.com/tools/repo
  git clone --bare $URL manifests.git
  mkdir -p manifests/.git; cd manifests/.git
  for i in ../../manifests.git/*; do ln -s $ı .; done
  cd ..
  git checkout $BRANCH -- .
  cd ..
  ln -s manifests/$MANIFEST manifest.xml  
You can trace what really happens with repo --trace init ...
Then, repo sync clones git repositories to .repo/projects for each project in manifest.xml and local_manifest.xml, creates working directories with .git having symlinks to the corresponding bare repository, checks out the branch specified in the manifest, and updates .repo/project.list.  The case where the projects are already there is slightly different, essentially performing a git pull --rebase.