A bag-of-words-model (BOW) is usually used to represent a text: you throw all the words together (as if in a bag), without keeping track of their sequence. This is a gross simplification over a text, as word sequencing plays an important role in creating the meaning of a text. But on the positive side it's easier to handle, eg in information retrieval tasks, where you might not need the precise meaning anyway.
So the BOW is discrete and symbolic, as it represents each of its elements by a set of words that are contained in it. Nothing numeric in there. You'd calculate the similarity of two items by comparing the two sets, how big is their intersection, and the difference between the two.
A CBOW is a slight modification: instead of the words, we use vector representations of them; and instead of having $n$ vectors for the $n$ surrounding words, they're all added up (formula 14.14) It's still a BOW, as the set of words used to represent an element is now the set of words surrounding it within a certain distance ($h$). What makes it continuous is the switch from a set of words (ie symbols) to a vector.
He contrasts this with a recurrent neural network, where words are represented by a state vector which gets updated after every new word, going back to the very beginning of the text. This would give different representations for the same word occurring in the same localised context, whereas the CBOW would return the same representation.
For example, for $h$ being 1 (to keep it simple):
when a word has a meaning, then a word has a purpose.
Now imagine we're interested in the encoding of word: in the recurrent case the first one is
when + a + word, whereas the second one is
when + a + word + has + a + meaning + , + then + a +word — the sequences here represent the updated state of the network after the respective words have been added.
In the CBOW case, both occurrences of word are encoded by
a + word + has (the word plus/minus one word either side, as $h$ is 1). So they will be identical.
To answer your question, continuous here is in contrast to discrete or symbolic, and indeed refers to a numerical vector.