Rat Man's dens are secret rooms hidden throughout Portal 2. They usually feature weird drawings on the wall, cans of open food, and radios on the ground.
Chapter 1 - Test Chamber 4 - After you fall through the glass, there is a staircase leading up to it.
Chapter 2 - Test Chamber 3 - This one is very easy to find, it's just a room off the main test chamber, no portals needed.
Chapter 2 - Test Chamber 6 - This one gets you the Final Transmission Achievement if you bring the radio from the garbage that GLaDOS launches at you. To get to it you must place a portal on the most upper right panel on the wall to the left of the two angled panels. Place the second portal on the right panel of the two angled panels. Launch yourself. The momentum should lanch you into the Rat Man's Den which is through a glass window on the wall to the right of the two angled panels.
Chapter 3 - Test Chamber 11 - there is a wall where the hard light surface hits it. Behind this wall is a room where you see Wheatley briefly and some text on the wall. It is open with a space big enough for you to go through on the side, but it is easy to not notice.
Chapter 3 - Test Chamber 12 - You'll see an enterance near the top of the chamber. Use the light bridge to enter it.
Chapter 3 - Test Chamber 16 - When you first walk in, there is a turret behind a vent. Use the cube to redirect the laser into that vent to kill the turret and open up the Rat Man's den.
Chapter 3 - Test Chamber 17 - Above the archway towards the exit, shoot the portal above that as far left as you can. Walk across the bridge and you should see an open panel on your right.