Initializing a new app requires you to mount the target directory into the container like so: docker run -v /my-new-cdk8s-project:/files brennerm/cdk8s init typescript-app.
Synthesizing works similar but instead of mounting an empty directory you need to mount the directory containing your cdk8s code. This could look like: docker run -v /my-existing-cdk8s-project:/files brennerm/cdk8s synth.
To be able to synthesize Python cdk8s code you need to use the brennerm/cdk8s:python image. It contains additional dependencies like Python 3.7 and pipenv.
No matter if you are initializing a new app or synthesizing your manifests, all generated files will have a root ownership afterwards. This happens cause cdk8s is being executed as root to be able to read and write files from and to the volume due to this issue. To fix this you can pass the -u uid=$(id -u) parameter to the docker run command. Tried to prevent executing as root but currently it does not seem possible while keeping a good user experience of the resulting image.