How does budding happen in roses even after the aplical meristem is cut off?

The phenomenon involved here is apical dominance which means the inhibition of the growth of lateral buds by the terminal bud of a plant shoot.

The apical bud (or tip) produces the growth hormone auxin, which not only promotes cell division, but also diffuses downwards and inhibits the development of lateral bud growth which would otherwise compete with the apical tip for light and nutrients. Removing the apical tip and its suppressive hormone allows the lower dormant lateral buds to develop and the buds between the leaf stalk and stem to produce new shoots which compete to become the lead growth. Thus removing the apical tip will prompt elongation of the secondary growing points which were previously dormant/surpressed.