What does 'in-trim speed' mean with respect to Boeing 737 mis-trimming correction?
"In-trim" speed means whatever speed the airplane will naturally seek if you let go of the controls.  If you set the stab to position X, the airplane will naturally seek speed Y, for a given weight, thrust, flaps, and C of G condition.  

So say the stab is at position X and this makes the airplane want to pitch to 240kt.  It will seek and maintain that speed if you don't touch the controls.  If you want to fly slower, you can add up elevator by pulling the column back, increasing the tail's downforce and pitching the nose up, until it slows to a speed where the forces are back in balance with the elevator held fixed at that position.  Say it's 200kt.  

You are now 40 kt below trim speed, and having to hold the elevator displaced to keep it there.  If you want to be able to let it fly hands off at 200kt, you have to move the stab itself so that the tail's downforce with the elevator at neutral is the same as when the elevator was being displaced.  So you trim out the elevator displacement (by applying trim and easing the column back to neutral in little trial and error inputs) and now you're in trim at 200kt, hands off.

The 737 has a stab screw jack with a manual operation feature with a cable circuit between the screw jack and the trim wheels in the cockpit that let the crew turn the jack directly with the trim wheel, with no electric input, when in manual trim mode.

If you are way out of trim, that is, holding a lot of elevator to make the airplane fly slower or faster than its trim speed, it puts a large rotational moment within the horizontal tail which increases the up or down load on the screw jack itself.  These jacks are usually acme screws with dual load path threads that have a fair bit of internal friction under load, and the effort to drive the screw when it's highly loaded can be more than you can apply with your hand on the little wheel on the sides of the center console.

So say you're in manual trim and the airplane is trimmed to 280 kt and you want to slow to 200kt, and pull the nose up with elevator to slow down and you don't retrim as you're slowing down.  At some point, if the out of trim condition and resulting the load on the screwjack is high enough, you can't move the trim wheel because you can't apply enough force to turn the jackscrew.  To avoid this, you should be retrimming the stab as you slow down.

Their procedure is to try to offload some of the out of trim load on the stab jack by moving the elevators back toward the original trim speed.  If you were holding the column way back to slow to 200kt while the airplane was still trimmed for 280kt, you'd release some of the pull, letting the airplane pitch over a bit and accelerate, while trying to move the trim wheel.  At some point you'll have unloaded the stab screw jack enough to make the wheel move.  You don't want to pitch over any more than you have to so you just relieve the elevator enough to get the wheel to move, and keep working it along until it works normally. 

In this context, "extreme cases" refers to the case of a large difference between existing speed created by elevator input and existing trim speed.