The fungus causing rose rust is, like all rusts, a biotroph: it infects the host tissues for extended periods without killing them, feeding on the living cells. Like all rusts, it is not able to survive on dead plant material, so must either alternate with a different, perennial host, or produce a resting spore to pass the dormant season.
Phragmidium tuberculatum and several other very similar species which infect roses do not have an alternate host; that is, they only attack roses and pass the winter as resting spores.
The first formed spores (spring spores) infect young stems, causing distortion and the production of bright orange pustules. These in turn infect the leaves to produce dusty orange spores (summer spores) which are spread by wind and initiate further infections. In late summer, the pustules producing summer spores switch over to produce the dark, tough resting spores. These spores survive the winter often adhering to stems or trellises. And then the infection starts over again in spring.
Infections may be severe enough to cause serious damage, but this is relatively rare and most infections are light enough not to require control.