Mathematically, the plane wave extends out to infinity in all its directions, and, of course, physically that cannot be I am currently studying Optics, fifth edition, by Hecht. In chapter 2.7 Plane Waves, the author says the following:

Mathematically, the plane wave extends out to infinity in all its directions, and, of course, physically that cannot be. A real "plane wave" is a finite thing that, no matter how big, only resembles a mathematical plane. Since lenses and mirrors and laser beams are all finite, that "resemblance" is usually good enough.

I don't understand what is meant here. Why is this not physically possible? And what do lenses, mirrors, and laser beams all being finite have to do with making the plane wave "resemble" being a plane? 
I would greatly appreciate it if people would please take the time to explain this statement.
 A: 
Why is this not physically possible?

Assuming that he universe is finite a true plane wave (which is infinite) is impossible. A second argument is, that a plane wave would require an infinite amount of energy and would neither have a starting time nor a end time. 

what do lenses, mirrors, and laser beams all being finite have to do
  with making the plane wave "resemble" being a plane?

Suppose a true plane wave would exist. If we would use a mirror of finite dimension to reflect the wave, the mirror would "destroy" the true plane wave. Hence, once we would use an optical element, the plane wave would no longer exist. 
However, since the laser beam itself is finite, the error induced by finite optical elements is usually "small". So the question is: Does a finite laser beam "resemble" a plane wave? Since many properties are only "mildly affected" by the finite dimension of the wave, the answer is YES. For example, let's consider the interference patter of two laser beams: In the region of the overlap of the two beams the interference pattern resemble the pattern expected by two plane waves. 
