Babies can develop thrush after coming into contact with yeast from other individuals. Thrush occurs when factors promote yeast overgrowth, leading to infection in babies. Signs of oral thrush include white patches in the mouth, while thrush in the nappy area appears as bright red patches with defined edges. Treatment involves boiling babies' items after use, rinsing their mouth with water after feeding, and treating breastfeeding mothers' nipples to prevent reinfection.