One-class classification may at first seem less useful than a more general approach that allows you to identify many classes. However, it can be quite useful, and in fact more difficult at times. A snippit of introduction motivating it is here on wikipedia, with some links to sources.
Although there are many instances where it is useful, I will stick to one example to try and help your intuition. Suppose you have some real world data generating process that you think will produce outliers at some time points. In a one-class classification, your goal would be to identify whether the new observation you observe is an outlier or not. If it is, you might take some sort of action.
Now, I'll dig a little bit deeper into why this is one-class classification. First a definition: An outlier is an observation that does not come from the distribution of the one-class in your dataset. Notice, this does not impose a structure on the distributions of the outliers. Each one can come from a distribution if you like. We only assume that the good (one-class) observations all come from the same distribution. In this sense, it really may not make sense to try and classify the outliers - they could be anything - but if we get a good idea about the one-class that we are interested in, this can suffice to help us identify and exclude outliers that we encounter.
Control charts are another where this type of approach is widely used.