What does the phrase "Simple For Loops" mean in computability theory?
The Wikipedia statement is informal and quite ambiguous. For example, let $A(n,n)$ be the Ackermann function, and consider the following program, where $n$ is the input:

x ← 0
for i from 1 to A(n,n):
    x ← x + 1
return x


This function is not primitive recursive, although there is a bound on the number of iterations which is known ahead of time.

A better explanation is the LOOP programming languages. Every loop in LOOP runs $x_i$ times, where $x_i$ is a variable, and the number of iterations is the value of $x_i$ before the loop is run. For example:

  LOOP x DO
      x ← x + 1
  END


is a problem that doubles $x$, and

  z ← 0
  LOOP x DO
      z ← z + 1
  END
  LOOP y DO
      z ← z + 1
  END


is a problem that assigns $x + y$ to $z$.

A function can be computed in LOOP (using a reasonable input/output convention) iff it is primitive recursive. So if you only allow loops in which the number of iterations is the value of some variable just before the loop, then the resulting function will be primitive recursive, and furthermore, every primitive recursive function can be computed only using such loops.