The typical logistic regression model is written as something like
where we model the log-odds by a linear combination of our predictor variables $x$. In the equation above $\pi$ would be the probability that you are interested in calculating a confidence interval for.
Now, rearranging terms, we know that we can estimate the probability $\pi$ as
where $\hat\beta$ are the estimated coefficents from your linear regression.
It should be noted that, since maximum likelihood estimates
are invariant to transformation, $\hat\pi$ may also be considered the
maximum likelihood estimate of $\pi$.
So now, construction of confidence interval proceeds using the fact that
We can then construct a $(1-\alpha)$ confidence interval for $x^T\beta$ as
and thus, finally, a $(1-\alpha)$ confidence interval for $\pi$ is therefore
Some of the details above I have excluded assuming that you understand how a logistic regression works in general, and its corresponding design matrix, etc.