How does instantaneous velocity equal average velocity in constant acceleration?

Consider the case of a particle accelerating at a constant rate of 1 meter a second per second from rest. The instantaneous velocity at time 0 is zero meters per second. After 100 seconds, the instantaneous velocity is 100 meters per second, neither of which has any relationship with the average.

Which is to say you appear to be right, but perhaps you - or I - have  misunderstood the formulation. Maybe they intended to say or print the much more comprehensible "average acceleration", and not "average velocity".

When the acceleration is zero then the velocity-time graph is a horizontal line, so there is no change in velocity and the average velocity will be equal to instantenous velocity.
Similarly when there is constant accelaration, the velocity time graph is also linear but now this time it is a line with the slope equal to the acceleration. Now if we draw a displacement time graph, then for a particular time interval the average velocity is the slope of the line joining the two points(say A & B) of the time interval. Here as the acceleration is constant, if we take a mid point(say C) on the graph, then on the lower part(say AC) the slope at any point(tangent) will be less than the slope of AB. But slope on any point on CB is more than AB. Meaning slope at C will be equal to AB. Now if we take the interval of A and B very very less then we will gradually reach to the point C. Thats why limit to zero of the of the interval of average velocity is equal to the instanteneous velocity at the point.