How did a one-line paradox break the foundations of mathematics?

Because it showed that the most natural definition of a set, any collection you can describe, leads to a contradiction. In 1901 Bertrand Russell published a set-theoretic paradox: it shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. That principle says that for any sufficiently well-defined property, there is the set of all and only the objects that have it. It sounds harmless.

Here is the paradox. Call a set normal if it is not a member of itself, and abnormal if it is. The set of all squares in a plane is normal, since it is not itself a square. Now take R, the set of all sets that are not members of themselves. If R is not a member of itself, then by its own definition it is a member of itself. If it is a member of itself, then it is not. That is Russell's paradox.

Why was that so serious? In classical logic any proposition can be proved from a contradiction, so a contradiction in a set theory destroys the conventional meaning of truth and falsity. Set theory was seen as the basis for all the other branches of mathematics, so the paradox threatened the foundations of mathematics as a whole. It also hit Gottlob Frege, whose system Russell showed could produce the paradox, undermining Frege's attempt to reduce mathematics to logic.

The repair came in 1908. Zermelo restricted the unlimited comprehension principle, and Russell proposed his own type theory. The standard theory that grew from Zermelo's, ZFC, does not assume that every property gives a set: it only lets you carve out definable subsets of a set you already have. The Russell set cannot be built that way, so in ZFC it is not a set.