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

Practical reasons

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

Computations in local cohomology

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.

Duality

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.

Vanishing and exactness

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 $\text{Hom}_R(I, \omega_R) = \omega_R(I^{-1}) \to \omega_{R/I}$. This is surjective because the next term is zero 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.

Ubiquity of Cohen-Macaulay rings

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.

Summands of regular (or Cohen-Macaulay) rings

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.

Complete intersections

Complete intersection rings are Cohen-Macaulay.

Rational/log terminal/F-regular singularities

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).

Pithy quotes

"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 ]