Nice looking logo. Especially the typography. Good work.

Now the process you are looking to perform is called a boolean process. (Some people call it "welding") In Illy it is performed with the Pathfinder tool. Your version is newer than mine, so the following might vary slightly.

1 Bring up the Pathfinder Palette Shift+F9 or Select it "Window>Pathfinder>"

2. Select your "f" and the white tails then choose "add" from the pathfinder tool. You may want to click the "expand button" after this too to visually connect the shapes, but if you don't expand them they remain "editable" in their original state.

3. Select your newly created "f" shape and the background portion you want to cut it out of, and use "subtract" this time.

4. If you need to cut the "f" out of both the red and the grey, make a copy of the "f" first (before step 3) so you have a duplicate to perform the boolean subtract a second time.

From the looks of that image, the 'tails' don't exist.
What I'd do is
1. select the red and gray shapes.
2. Create compound path (make sure the F stays in front)
3 SElect the F and also select your compound red/gray thing.
4. Use Minus Front in the Pathfinder palette.
done

You know, I think your right PrintDriver! Good eye. I thought it looked like some little rectangles laid over top.