The Rules And Tricks Use To Find The Equivalent Resistance
But what is the basic of folding it on that line?

The rule is simple. If the circuit is symmetrical then we can fold on the line of symmetry. In the original circuit the voltages on each of the mirrored node pairs would be identical so we can use one set of resistors but at twice the current so half the resistance (R/2).
Note that the resistors on the axis of symmetry did not change their value.

What are the requirements for folding the circuit for simplify?

Only that it is symmetrical.

How is it not differ the initial circuit after folding on that line?

It is different but it is equivalent. Internally you have changed the circuit but when looked at from the outside (the left wire and right wire) you can't tell the difference. If you measured the resistance between the two outer terminals it is the same in both cases.