well are you sitting comfortably.... this may take some time.

there are several triggers from day length/light quality, temperature and plant hormones and they may all interacting.

The commonest is light quality [wavelength of the light]/day length, this triggers the production of plant hormones that then trigger cells to form buds. As you keep dead heading, the plant doesnt get another hormone produced from the developing seeds that supresses flowering in favour of seed production. So you get more flowers.
Temperature in some plants also triggers hormone production but I'm not certain about roses.

Hope this explains a very complex bit of biology.

To make it even more complicated, different types of roses respond to different combinations of cues. Most species roses, many old hybrids, some climbers, and some modern shrub roses form flower buds in declining hours of daylight (i.e., in fall) but won't open the buds until after a period of cold weather, and in increasing daylight. Most "everblooming" modern roses form buds after the hormones in the growing tip have indicated that a certain number of leaves have formed--in modern climbers, that number is pretty high in upright stems from the base, but not in the side stems from those. All of this is also dependent on the bush having enough energy from sunlight to be able to budget in flower production.

My goodness, a simple question that prompted complicated answers! I'm much obliged to you both for taking the trouble to explain the astonishingly complicated processes that dictate the behaviour of the plants.

Complex indeed, and even those far more expert than us really don't understand it all!