A traditional model is given a fixed, human-chosen set of features. Once those features are in place, adding more examples mostly refines the weights attached to them; the model cannot invent a new property of the data that nobody thought to encode. Its accuracy therefore tends to rise quickly with the first batches of data and then flatten, because the ceiling is set by how informative the chosen features are.
A deep model is different in kind, not just in degree. Its layers contain a large number of adjustable parameters, and those parameters are not specified by a person; they are pinned down by examples. With too few examples, many different parameter settings fit the training data equally well, and the model has no basis for choosing the one that generalizes. As the dataset grows, the space of settings that fit the data shrinks toward the settings that actually capture the underlying structure, so accuracy keeps climbing well past the point where a traditional model has plateaued.
This is what 'data-hungry' means precisely: it is not that deep learning refuses to run on small data, but that its advantage over traditional methods only appears once there is enough data to constrain its many parameters. On a small dataset the deep model is often the weaker of the two.