In the real world, Hamkins' program would search forever, never outputting anything. In the context of a "nonstandard" version of the line of natural numbers, "finite" no longer means what you think it means. In this context, Hamkin's program does output something after a "finite" number of steps, and Hamkins can make the program output whatever he wants by changing the meaning of "finite" instead of changing the program.

Universe means: model of set theory.

Goedel's incompleteness theorems guarantee the existence of non-standard models of set theory (and Peano arithmetic). Hamkins cleverly uses non-standard models to construct a single program that can (simplifying a bit) exhibit arbitrary behaviour, depending in which universe (= model of set theory) you interpret it.

This is just maths, there's nothing dry about it compared to any other article about maths.