> > Suddenly, the monstrosity of infinity, long feared by mathematicians, could no longer be relegated to some unreachable part of the number line. It hid within its every crevice.

Think of the number line stretching from negative infinity to positive infinity and let C represent the cardinality/size/count of numbers on that number line. Now just take portion of the number line from 0 to 1. Let C1 represent the cardinality/size/count numbers from the truncated line from 0 to 1. You would assume that C > C1. But in fact they are equal. There are just as many infinite real numbers from 0 to 1 as there are on the entire number line. Even worse, this hold true for any portion of the number line, how small or big you make the line. Rather than infinity being in a far distance place at the edge of the line in either direction, there is infinity everywhere along the number line.

> I don't get what "suddenly" became apparent.

It appeared suddenly because prior to cantor/dedekind, mathematics only understood the countably infinite ( natural numbers, integers, rationals, etc ) . By constructing a complete number line, cantor/dedekind showed there is a cardinality greater than infinity ( countable ). The continuum.

Cantor also showed that there is an infinite number of cardinalities.