How to extract the source files from a git repository?

The terminology git uses for this is clone and it means to make a copy of the repository in the current directory.

For example:

git clone https://github.com/path/to/project 

Optionally you can specify a target directory and put your project there like so

git clone https://github.com/path/to/project path/to/directory

From Git's documentation.

Your contractor left you a bare repository. It is used by Git for remotes that don't have a working copy (for example, on a server).

Just clone from the bare repository:

git clone project.git

You should end up with a directory called project that has a checkout of the master branch and a .git directory with a clone of the repo. This project directory is a full copy of the repo, but it's the kind you can do useful things with.