How to draw cement-wall-like transition background in Adobe Illustrator?

Textures like these are mostly bitmaps and, hence, better done with Photoshop or another bitmap editor.

Creating textures is possible within Illustrator using the different Photoshop effects in the Effect menu. But, as the name suggests, these are not vector effects. And it’s not easy to achieve the effect you want if it isn’t in the default effects.

A way better method of using textures is to actually obtain pixel images, edit them in Photoshop and then place (or paste) them into Illustrator. You can shoot the images yourself as photographs, or rely on stock images. One image bank I keep on returning to is CGtextures, which has a huge collection of royalty-free images originally meant for use in constructing 3D environments.

In the case you give, you’d want to add to add a gradient to the texture. This can be done by creating an object on top of the texture, giving that a gradient and setting its blending mode to Myltiply, Color Burn or something similar.