1. LDA or logistic regression?
LDA and logistic regression can both be used to 'predict' the class of a subject, both can handle the case of more than two classes.
They both differ in the way of solving the classification problem and therefore they make different assumptions: logistic regression assumes the well-known S-shape, while LDA assumes that in each class your data are (1) multivariate normal and (2) with the same var-covar matrix in each class.
If the assumptions of multivariate normality and same var-covar are fulfilled then, in general, LDA will perform better.
2. Intuition behind LDA
You have 'subjects' that are characterized by features $x_1, x_2, \dots x_n$. The goal is to decide on the class of the subject, knowing the value of its features.
As said, LDA assumes that, in each class '$c$', your features have a multivariate distribution with a mean that depends on the class, so $\mu_c$ (note that this is a vector) and var-covar $\Sigma$ ( the same for all the classes), so for each class we know the multivariate density $\Phi_c(\mu_c,\Sigma)$ that allows us to calculate the probabilities.
Now, given the features, we can compute the $\Phi_c$ for the featurs $x_i$ and we will put the subject in that class $c$ where this yields the highest value (i.e. Where the 'probability' is heighest).
3. Why is LDA a dimension reduction technique?
LDA assumes multivariate normality in each class with the same var-covar. Therefore the classes are all the 'same' except for their mean. So you can 'feel' that the number of means will be important.
If you have $n$ features, then, in the end, the solution will depend on $C$ means, $C$ being the number of classes. In fact it can be show mathematically that LDA 'solves' the classification problem in a 'subspace' of the $n$-dimensional feature space, and that this subspace has dimension that is lower than $C$.
To make it more concrete, assume that you have subjects with $25$ features and you want to classify them in two classes, then you can 'solve' the problem in a one-dimesnional space (thus on a line). This is why LDA is said to be a dimension reduction technique, in this case it reduces the dimension from $25$ to one .