I cannot understand how we arrived to these bounds... the article says z equals to two expressions and then somehow we get bounds where z is in between them. I do not understand why plane is upper bound and paraboloid is lower bound... (link to article https://www.khanacademy.org/math/multivariable-calculus/integrating-multivariable-functions/modal/a/triple-integrals)
Consider the region R bounded by the following two surfaces:
The paraboloid $z = x^2 + y^2$
The plane $z = 2(x + y + 1)$
To figure out the region is a good start try to plot it in the planes $z-x$ or $z-y$. In this way you can easily convince yourself that there is a region where the plane is upper bound and paraboloid is lower bound.
To find this region region in the plane x-y you can equate the equation of the paraboloid and of the plane and find that it is a circle, indeed
$$x^2+y^2=2(x+y+1)\implies (x-1)^2+(y-1)^2=4$$
Note that from the inequality
$$2(x+y+1)>x^2+y^2\implies (x-1)^2+(y-1)^2<4$$
we can directly deduce that the region where the plane is upper bound and paraboloid is lower bound corresponds to the interior of the circle.