There are two main ways in which recycling can affect the environment. The first has to do with the use of resources and the second has to do with how things are disposed of if they are not recycled.
Everything that we use is made of resources. Paper comes from trees, aluminum and "tin" cans come from ores that have to be dug out of the earth. When we use these resources and then throw them away, they are lost. If we recycle them, we do not have to use new resources (cut down new trees, dig new ores). This means there will be more resources for longer.
If we don't recycle, we have to simply throw things away. Typically, this means that they will go into landfills. In the United States, at least, landfill space is relatively scarce. The more we recycle, the less land we have to waste creating landfills.