Intuition for the CES consumption index in New-Keynesian DSGE models
Heuristically, you can think of the integral as just a sum:

$$ \bar{C} = \left( \sum_{i=1}^n C_i^{1-\frac{1}{\epsilon}} \right)^{\frac{\epsilon}{\epsilon - 1}} $$

where $\bar{C}$ is an index of aggregate consumption, and utility is given by $u \left( \bar{C} \right)$.

It's easy to check that the marginal rate of substitution between goods $j$ and $k$ is given by

$$ \text{MRS}_{jk} = \left( \frac{C_j}{C_k} \right)^{-\frac{1}{\epsilon}} $$

which means that the (absolute value of the) elasticity of substitution is $\epsilon$.

With a continuum of goods, the analogue of summation is integration, and you can (loosely) think of the MRS between goods $j$ and $k$ as

$$ \left( \frac{C_t (j)}{C_t (k)} \right)^{- \frac{1}{\epsilon}} $$