Many images which use textures or "grunge" appearances in Illustrator take advantage of Opacity Masks and embedded raster images. Even though Illustrator is vector at its core, there are times where utilizing the subtle variations in a raster image can be helpful. Trying to create distressed or mottled backgrounds is certainly one such case.
First, find a raster image with a texture you like. It's easiest if it's a greyscale image (but color images work). Greyscale simply allows the raster image to adjust values rather than possibly also altering the color of (what will be) underlying objects.
So a grey scale texture image:

Open the Illustrator artwork:

Now, if I want to add the texture to the light brown background area:
- Import the raster texture image
- Position it above the object I want to apply it to
- Select the texture image and the object below it
- Click the
Make Mask
button on the Transparency Panel
You can then click the mask thumbnail on the Transparency Panel and move the raster image around to see how it effects the object it is applied to.
Large animated gif attached. It may take a moment to load

What the Opacity Mask does is hide portions of the light brown rectangle allowing the dark brown rectangle to show through it. Like all masks, the dark areas of the raster image hide content and the white areas show content.
You could apply an Opacity Mask to the artwork as a whole or use several varying textures and apply them to separate objects.
This is often how "grudge" appearances are achieved with Illustrator.
Be aware: This does use raster images. And with that in mind the raster mask is just as susceptible to "broken pixels" as any raster image should to overly enlarge the artwork. Unlike most vector creation, you need to pay attention to the PPI and any scaling you do if using this technique.
You could also auto-trace the raster texture and then use a standard clipping mask in Illustrator. However, often auto-tracing raster texture images results in unwieldy large files which can quickly become unworkable. Illustrator tends to get slower and slower with each new complex vector object. In addition, some subtle texture variations simply aren't possible using vector paths.
The following existing questions also refer to how to "distress" vector graphics:
Add imperfections to vector graphics / drawings
How to get a construction paper look in photoshop or illustrator