Leeching is basically loss of nutrient from soil. Normally nutrient is rained into the soil; absorbed by plants; shed in their leaves/when they die; digested by decomposers so its back in the soil. If you take away the vegetation you remove nutrients from the cycle. In addition to this the soil is not protected by plants and so when it rains there will be a higher rate of surface run off, this will take the nutrients from the soil with it. Soil erosion is also caused by the fact that without plants to protect the soil there is more surface run off, because soil is taken with it.
Plants absorb water from the soil and lose water from their leaves (through transpiration) in to the atmosphere which goes on to make clouds. If there are less plants then less water is evaporated into the atmosphere, this means there are less clouds; less clouds means less rain, which can mean drought.
Plants also convert carbon dioxide into oxygen when they photosynthesise. Forests carbon sinks, they use more carbon than they release: this means they help to make sure there aren't too high levels of CO2 in the atmosphere. When forests are cut down this process is lost and additionally the trees are usually burnt which releases CO2 into the atmosphere.
This article is explains some of the effects of deforestation nicely: