I think there are many reasons.  Here are a few.

<h1>Practical reasons</h1>

Cohen-Macaulay rings are just plain easier to work with.  

<h3>Computations in local cohomology</h3>
For example, any number of computations in local cohomology modules become much easier in the Cohen-Macaulay case (see for example Bruns and Herzog's book on the topic).  Explicitly, it's much easier to determine if a class in $z \in H^{\dim R}_{\mathfrak{m}}(R)$ is zero or not in the case that $R$ is Cohen-Macaulay.

<h3>Duality</h3>
Both Grothendieck-local and Grothendieck-Serre duality work much better in Cohen-Macaulay rings.  The dualizing complex (assuming it exists) is a complex whose first non-zero cohomology is the canonical module and which is equal to this (shifted) canonical module if and only if the ring is Cohen-Macaulay.  Without this hypothesis one frequently needs to work in the derived category and do numerous computations with spectral sequences.  It is convenient to not have to.

<h3>Vanishing and exactness</h3>
If $R$ is Cohen-Macaulay and $I$ is a height-one ideal (and suppose the rings are quotients of Gorenstein/regular rings so they have dualizing complexes).  Then we have a *surjection* of canonical modules $\omega_R \to \omega_{R/I}$.  This is surjective because the next term is zero precisely when $R$ is Cohen-Macaulay. This sort of vanishing applies to more general situations and is really useful (there is a local dual version involving local cohomology).  There are lots of other vanishing results that you can deduce from this kind too.


<h1>Ubiquity of Cohen-Macaulay rings</h1>

Ok, if Cohen-Macaulay rings weren't so common, the above nice properties would be less interesting.  But Cohen-Macaulay rings are really common.  Here are some examples.

<h3>Summands of regular (or Cohen-Macaulay) rings</h3>
If $R \subseteq S$ is a extension of rings and $R \to S$ splits as a map of $R$-modules, then if $S$ is Cohen-Macaulay, so is $R$ (the point is $H^i_m(R) \to H^i_{mS}(S)$ injects and the latter term is zero, at least after a little localization on $S$ if necessary).  Lots of rings coming from representation theory for instance are summands of regular rings.  

<h3>Complete intersections</h3>  
Complete intersection rings are Cohen-Macaulay.

<h3>Rational/log terminal/F-regular singularities</h3>  A lot of classes of singularities which are most useful today are Cohen-Macaulay.  One of their most useful properties is their vanishing properties (see above).

<h1>Pithy quotes</h1>

*"Life is really worth living in a Noetherian ring $R$ when all the local rings
have the property that every s.o.p. is an $R$-sequence. Such a ring is called
Cohen-Macaulay (C-M for short)."*

[Page 887 of Hochster, [Some applications of the Frobenius in characteristic 0][1] ]


  [1]: http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.bams/1183541144&page=record