A transfer between Cologne Bonn Airport and Stuttgart Airport covers 364 km in around 3h 45. It exists for the day the connection breaks — when the replacement flight leaves from a different airport and getting there is suddenly your problem rather than the airline's.
The pickup is driven by your inbound flight number: the driver watches the actual landing time and waits 60 minutes free after touchdown, which is what a diverted or delayed arrival needs. At the far end the drop-off is at the correct terminal for your departure — Stuttgart Airport has 4 and the wrong one costs a shuttle ride you do not have time for. Tell us the departing flight at booking and the schedule is worked backwards from its check-in deadline rather than forwards from a convenient departure. We run the same journey to Hamburg Airport, priced the same way.
Luggage stays in the car for the whole 3h 45, which is the single biggest practical difference from any rail route: nothing is lifted onto a train at Cologne Bonn Airport, off it again at the interchange, or onto a second one for the run into Stuttgart Airport. If the airline has rebooked you and is reimbursing ground transport, the invoice shows the route, the date and the fare in a form an airline will accept.