Why do the engines seem to power down and then back up after take-off?
There are several kinds of power reduction which could happen after takeoff:


Cutback (noise reduction when the airplane is very low, after takeoff)
Climb thrust


Both are set on the CDU (Control Display Unit, the screen and keyboard of the FMC, Flight Management Computer).

Since you describe a temporary and dramatical power decrease, this should be the cutback. When you hear the power increase again, it must be that the setting goes from cutback to climb thrust.

In the B737-800

The cutback option on the CDU


This is the CDU (Control Display Unit), where you set up the cutback. It is in the second page of TAKEOFF REF. Here is the screen before turning cutback ON.



ACCEL HT : This is the height (above ground level) at which we will let the plane accelerate (we will lower the pitch a little bit and gradually retract flaps).
EO ACCEL HT: Same as above, but applies when one engine is not running (Engine Out Accelaration Height). This setting will be used in case of engine failure after takeoff.
REDUCTION: This is the height at which the thrust setting will go from TOGA to CLIMB. We need maximal thrust during takeoff, because it's a critical phase and it's safer. But at the same time, keeping this setting too long decreases the engines lifetime because of the high temperature. It also burns more fuel, it's noisier and less environment-friendly. That's the reason why the airplane won't climb with maximum thrust.

But this question is about cutback. Here is CDU again after turning cutback ON.



CUTBACK N1: This will be the target N1 setting when the REDUCTION height will be reached. The Auto-throttle together with the EEC will arrange for maintaining this lower shaft speed (here you can see it's 84,6%). This will be a lot less noisy but will still provide enough power to climb!
EO ACCEL HT: Same as above
REDUCTION: At this height, cutback will happen and CUTBACK N1 will be the engines target N1 speed.
RESTORE: This is the height at which cutback is no longer needed, and climb thrust N1 setting will be the new N1 target for the engines (92.3% in this example).



Takeoff with cutback


This is the takeoff roll. IAS indicator is not active yet (below 45KIAS). Ground Speed is currently 35kt.
N1 and TO/GA are displayed on the FMA (Flight Mode Annunciator) at the top of the PFD (Primary Flight Display). On the right, you can see on the ED (Engine Display) that the green marker is indicating TO thrust (because of TO/GA mode).

Few seconds after the rotation. You can see at the bottom of the PFD the radio altimeter indicating 300ft AGL. We haven't reached the REDUCTION height, so the cutback is not active yet. The ED still indicates TO thrust and the green markers are just below 100% N1.

Few seconds after reaching REDUCTION height. N1 is active on the FMA. Radio altimeter indicates 960ft AGL, this is above our REDUCTION height, so the cutback should be active and our engines should be a little quieter. A look at the ED and we can see that the green target N1 marker is now indicating 82.6% N1. Our engines are still rotating faster than that (98% N1), and the EEC is currently decreasing the fuel flow arriving to the combustion chambers.

We are now reaching 2500ft AGL. This is below the RESTORE height of 3000ft, so the cutback is still active, as shown by the ED. The green target N1 marker is indicating 83.0% N1 and the current N1 for both engines is also 83.0% N1. In 500ft, the N1 target speed should increase again to climb thrust settings.

We've just reached the RESTORE height (above 3000ft AGL), and we can see the green markers show 92.3%. We are back to the climb thrust settings and our plane is loud again! The flaps are still down, and the pink speed marker on the PFD just moved to the top. We are going to retract the flaps and slats, and accelerate to 250KIAS. It seems that RESTORE height is also the ACCELERATION height when cutback mode has been turned on. [I'm not sure, could a B737 pilot confirm this?]

The flaps are up, our speed is now 250KIAS.



Further readings


http://www.boeing.com/commercial/aeromagazine/aero_21/quietclimb_story.html
http://www.flaps2approach.com/journal/2013/6/29/avoiding-confusion-using-the-following-acceleration-height-t.html