Why did he let (y) equals to (1/2 sec t) ? and how did (-1) & (- √2/2) become (2π/3) & (3π/4) ?

Just to go into more detail than I did in our PMs and the comments on your last question...

You have to keep in mind that the limits of integration, the interval , only apply to the original variable of integration (y).

When you make the substitution , you not only change the variable but also its domain. To find out what the new domain is is a matter of plugging in every value in the y-interval into the substitution relation to find the new t-interval domain for the new variable (t).

After replacing and the differential with the new variable and differential , you saw that you could reduce the integral to -1. This is a continuous function, so the new domain can be constructed just by considering the endpoints of the y-interval and transforming them into the t-domain.

When , you have .

When , you have .

Geometrically, this substitution allows you to transform the area as in the image below. Naturally it's a lot easier to find the area under the curve in the second graph than it is in the first.