So,the integral is equal to zero because the Q$Q$ that is enclosed in the Gaussian surface is zero.That does not necessarily mean that the electric field is equal to zero.The only other way for the integral to be equal to zero is if the sum of the dot products inside the integral is equal to zero.It means that you have equal negative dot products as positive ones,thus giving you the integral of l(hat).dS=0$\hat l\cdot d\vec S=0$ where l(hat)$\hat l$ is the direction of the electric field.
To intuitively think about it you need very good visualization skills.