I do not understand what it means in this context: does it mean I can choose to pay the taxi on a hourly rate instead of a flat fee, or instead of paying on the number of kilometers of the ride ?
If you click the button on a similar form I found, it brings up a choice of a number of hours, so my guess is that instead of charging you to go from one place to another place, you can reserve the car for a number of hours and it will take you multiple places. To "go by the hours" is not idiomatic.
What they probably meant was "I want to rent/reserve by the hour". For example,
With Legends Limousine, you can reserve comfort and style by the hour. (Source)
This means that you pay an hourly rate, or a fixed amount for each hour that you have the car service.