The common cold is exclusively caused by viruses. Antibiotics do not help fight it at all, your daughter's recovery was probably all her own strength :)

Antibiotics are frequently misunderstood and misused, so it's hard for us to know exactly what the logic was, short of telepathic power. Occasionally, antibiotics will be prescribed for cold and flu patients if the doctor is worried about secondary infections like bacterial pneumonia taking advantage of the weakened body in the meantime. There's also a lot of unfortunate cases where dodgy doctors prescribe them as placebo to basically just morale boost patients.