Upon contact with any of these three plants, the oil binds to your skin. The first time this happens, your body recognizes this "attack" and prepares an immune response. You seldom have a reaction to your first contact, but your immune system certainly does.
For your second encounter, your immune system is poised to launch a severe allergic reaction. When you brush against one of these nuisances again, the Urushiol oil triggers a powerful reaction.
Like a mosquito bite, but 1,000 times worse, your immune system tries to eliminate the areas of your skin that made contact with the plant. Those areas of your skin begin to itch like mad. Scratching the itch only makes it worse, because scratching increases your immune response, and can spread the Urushiol to unaffected areas.
At this point, you have a couple weeks of misery ahead of you, as your body gradually sheds the "contaminated" skin, and the allergic reaction runs its course. Calamine Lotion and anti-itch potions can help here. They help the healing process by numbing the itch reflex, and by providing a coating to your skin that supports the healing process.