Why consumption is defined as $ C_t:= \left(\int^1_0C_t(i)^{\frac{\epsilon-1}{\epsilon}}di\right)^{\frac{\epsilon}{\epsilon-1}} $
For some reason Ubiquitous edited this information into the question, rather posting it as an answer:


  $i$ indexes the goods available for consumption


Integrals are just sums with a lot of components (technically infinite). Here the space of goods is assumed to have large dimension, rather than just consisting of two or three types of goods.