After it rains, where does all the water from the puddles go? 


This is evaporation.

When it rains water collects in different places – in rivers, lakes, oceans and in puddles on the pavement. When the sun comes out it heats this water, turning it into water vapour or steam which then leaves the river, lake or ocean, goes into the air and rises high into the atmosphere (because warm air always rises).

Our clothes dryers, hair dryers and kettles just speed up the process of heating water and turning it into vapour or steam by applying more concentrated heat.