Hashing Trick - what actually happens
The a second bullet is the value in feature hashing.  Hashing and one hot encoding to sparse data saves space.  Depending on the hash algo you can have varying degrees of collisions which acts as a kind of dimensionality reduction.

Also, in the specific case of Kaggle feature hashing and one hot encoding help with feature expansion/engineering by taking all possible tuples (usually just second order but sometimes third) of features that are then hashed with collisions that explicitly create interactions that are often predictive whereas the individual features are not.

In most cases this technique combined with feature selection and elastic net regularization in LR acts very similar to a one hidden layer NN so it performs quite well in competitions.