I realize that these objects were originally created by Major Percy Macmahon and today have many applications but what was the original motivation for studying them?

MacMahon invented a technique which he called partition analysis to determine (multivariable) generating functions for many combinatorial objects and as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. This was introduced in his book "Combinatory analysis". Edit: I was a bit rushed to conclude that $\Omega$ had something to do with the motivation to look at plane partitions, see Richard Stanley's answer. I still believe that it was part of the machinery that he built for the same kind of problems that inspired looking at plane partitions. (I mean all of the results about counting tuples of integers satisfying sets of equalities/inequalities.) 


It does not seem from MacMahon's first mention of plane partitions that the $\Omega_\leq$ operator was relevant. At the end of Article 42 of his paper "Memoirs on the theory of the partitions of numbersPart I", MacMahon says "This partition may be termed 'graphically regularised' by reason of its origination in a subjacent succession of lines in the bipartite graph. This species of regularisation is the natural extension to three dimensions of Sylvester's graphical method in two dimensions." He then goes on to develop some simple properties of plane partitions (without using that terminology) and to conjecture his famous generating function $\prod_{n\geq 1}(1x^n)^{n}$. He also suggests less confidently that threedimension partitions have the generating function $\prod_{n\geq 1}(1x^n)^{{n+1\choose 2}}$ (now known to be false). The $\Omega_\leq$ operator is used implicitly to prove some simple results, but it does not seem to be relevant to MacMahon's original motivation. I believe that MacMahon did not explicitly use his $\Omega_\leq$ operator until "Memoirs on the theory of the partitions of numbersPart II", about three years after Part I. In Part II he does consider plane partitions as an example. 


Gjergji's answer, while it's a correct answer and a good answer, and the question itself only address the very first mention of plane partitions in the literature, and not the almostfirst mention that was clearly in the air at the time. Yes, MacMahon had in mind generating function theory. However, plane partitions are also a special case of semistandard tableaux, which are used to enumerate basis of irreducible representations of the general linear group. They are a variation of standard tableau, which enumerate bases of irreps of the symmetric group and were first described by Alfred Young a little bit before MacMahon's work. I would say that semistandard Young tableaux are more like a fellow traveler of plane partitions than an "application". MacMahon must have known something about it. (But conceivably some of these connections were only cleaned up decades later.) 

