Models at Dynamic Stochastic General Equilibrium level must be able to replicate real economies to an acceptable degree. One of the features of real economies has been a relatively stable growth rate (see also this post), $\dot x/x=\gamma$, where the dot above a variable denots the derivative with respect to time.
So one would want a model that admits a constant growth rate at its steady-state. In the benchmark deterministic/continuous time "representative household" model, the Euler equation takes the form
$$r = \rho - \left(\frac {u''(c)\cdot c}{u'(c)}\right)\cdot \frac {\dot c}{c}$$
This is the optimal rule for the growth rate of consumption. The rate of pure time preference $\rho$ is assumed constant. The interest rate $r$ has its own way to become constant at the steady state. So in order to obtain a constant consumption growth rate at the steady state, we want the term
$$\left(\frac {u''(c)\cdot c}{u'(c)}\right)$$
to be constant too. The Constant Relative Risk Aversion (CRRA) utility function satisfies exactly this requirement:
$$u(c) = \frac {c^{1-\sigma}}{1-\sigma} \Rightarrow u'(c) = c^{-\sigma} \Rightarrow u''(c) = -\sigma c^{-\sigma-1}$$
So
$$\frac {u''(c)\cdot c}{u'(c)} = \frac {-\sigma c^{-\sigma-1} \cdot c}{c^{-\sigma}} = -\sigma $$
and the Euler equation becomes
$$\frac {\dot c}{c} = (1/\sigma)\cdot (r-\rho)$$
Barro & Sala-i-Martin (2004, 2n ed.), extend the required form of the utility function when there is also leisure-labor choice (ch. 9 pp 427-428).
These fundamental property extends to the case of stochastic/discrete time.
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To compare, if we have specified a Constant Absolute Risk Aversion (CARA) form, we would have
$$u(c) = -\alpha^{-1}e^{-\alpha c} \Rightarrow u'(c) = e^{-\alpha c}\Rightarrow u''(c) = -\alpha e^{-\alpha c}$$ and the Euler equation would become
$$\dot c = (1/\alpha)\cdot (r-\rho)$$
i.e.here we would obtain a constant steady-state growth in the level of consumption (and so a diminishing growth rate).