What causes the upward motion in a nutating top? 
The locus of the rotational axis of a symmetrical rotating top with a fixed base is shown. This shows a nutation bounded by two circles.
What is the intuitive explanation as to why the top axis turns upwards at the lower circle and moves up?
 A: For the intuitive answer: see my 2012 discussion here on stackexchange of gyroscopic precession
As we know: while purely precessing is a solution to the  equation of motion (of the spinning top), the general solution is a combination of precessing motion and nutation.
(In classroom demonstrations the gyro wheel is usually released gingerly. That way of releasing the gyro wheel  tends to selectively suppress the nutation.)
The 2012 answer gives an intuitive explanation of gyroscopic precession by showing that a spinning object responds in a particular way to change of orientation of the spin axis. In response to change of the orientation of the spin axis the gyro wheel's spin axis tends to turns around an axis perpendicular to the existing change of orientation. Specific to nutation: as a consequence of being perpendicular there is no overall acceleration. (Compare the case of uniform circular motion. There is a force, causing acceleration, but it acts perpendicular to the existing velocity. As a consequence the angular velocity of the circular motion does not change.)
For the purpose of exposition we can conceptually divide the nutation motion into 4 quadrants:
-from downward to horizontal
-from horizontal to upward
-from upward to horizontal
-from horizontal to downward
downward to horizontal
In response to the downward change of orientation of the spin axis the spin axis shifts to horizontal change of orientation.
horizontal to upward
After having been shifted from downward to horizontal the spin axis is changing orientation faster than would be the case in pure precessing motion. The surplus then transforms from horizontal to upward.
upward to horizontal
During the upward stage the change of orientation of the spin axis is such that the ensuing horizontal motion is in the opposite direction of the overall precessing motion. Depending on the amplitude of the nutation the highest point of the nutation can come out as a cusp.

The key point:
The orientation of the spin axis has a rate of change. There is an acceleration effect that acts perpendicular to the existing rate of change. Since the effect is perpendicular: the overall rate of nutation remains the same.
See also:
Svilen Kostov and Daniel Hammer did a tabletop experiment to confirm that nutation behaves in accordance with the theoretical prediction. The title of their article is "It has to go down a little, in order to go around"
