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
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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.