For the first empty box, it refers to the formula just before it:
DE² + EF² = DF²
This formula is the Pythagorean theorem, which can be used in a right triangle to calculate a relation among the legs and the hypotenuse.
For the second empty box, he used the formula from the Pythagorean theorem and substituted some values, since DE = AB and EF = BC.
This substitution made is called the substitution property of equality.
For the third empty box, he wrote that DF² = AC², since the left sides of the equations DE² + EF² = DF² and DE² + EF² = AC² are the same, so the right side also are equal. This property is also the substitution property of equality.
For the fourth empty box, he already has all three pairs of sides of each triangle congruent, so the triangles can be proven congruent applying the SSS criterion. If all corresponding sides of two triangles are congruent, so the triangles are congruent.