There are specialized carpet cleaning products for this that are widely available in stores. For the ones that I am familiar with, you spray it onto the dirty area, brush it into the carpet, let it dry, then vacuum it up.

There are larger machines that one can rent for "shampooing" the entire carpet area, which is something that ought to be done regularly. There are also professional services that do this.

I'd rather just have a rug that you can take outside and beat the dirt out of.