On a college application how do I show that I do volunteer work?

If you're applying through the Common App, you will only have to write about your work in the activities section. That should be enough. If you wish to talk about it in more detail, you can do so in the "Additional information" section or in your essays. Most colleges will believe you and won't ask for any further proof. You may also collect recommendation letters from people you've worked for: that will make your contribution more evident.