When you present reasons why you believe in something, you argue for it. For example, writing an editorial to your local newspaper is a great way to argue for the creation of a dog park in your town.
You may think of argue as what happens when angry people shout at one another, but that is only one meaning of this verb. The other is "to present reasons and evidence about a stance or opinion." For example, if you give a speech in support of world peace, you might argue that nations accomplish more when they work together, giving examples to support this idea.
