Contrary to its name, logistic regression is a classification method. It is a very powerful one when it comes to text-based classification; it achieves this by first doing a regression on a logistic function, hence the name.
To get an initial understanding of the way logistic regression works, let's first take a look at the following example where we have artificial feature values X plotted with the corresponding classes, 0 or 1. As we can see, the data is noisy such that classes overlap in the feature value range between 1 and 6. Therefore, it is better to not directly model the discrete classes, but rather the probability that a feature value belongs to class 1, P(X). Once we possess such a model, we could then predict class 1 if P(X)>0.5, and class 0 otherwise.
Mathematically, it is always difficult to model something that has a finite range, as is the case here with our discrete labels 0 and 1. We can, however, tweak the probabilities...