Confused physics question?

It is a sinusoidal motion similar to riding a bike.  The balance is unstable.  If the mass is not vertically above the basal area ( width of skate) the person would fall over.  But the curve of the skate causes the skate to move across the path and the person is alternately falling left then right.

So imagine the skate is a bit to the left of the centre of mass.  It leans to the right.  The curve of the skate causes it to curve to the right. It passes across the centre of gravity and now the skate is a bit to the right.  The person starts to fall to the left.  But that tilts the skate to the left and the curve causes the skate to move to the left.  Now the cycle starts all over again.

If the skater is a beginner then each bend is bigger than the one before and shortly the skater falls over.  If the skater is competent then the bends get less on each cycle.  Even if you give the skater a sideways push.  So the skater recovers from any bump or irregularity.