Just load the WorkflowPass compiler pass and the services.yaml in your project.
After that, when a class supports workflow it'll have two routes (assuming entity dummy):
api_dummies_state_item PATCH ANY ANY /api/dummies/{id}/state.{_format}
PATCH with data (must be state key) {"state": "state_name"} to change a given resource's state to state_name.
api_dummies_state_get_item GET ANY ANY /api/dummies/{id}/state.{_format}
GET receives available states for the given resource.