Where does the "deep learning needs big data" rule come from

The original slide in question "Scale drives deep learning progress" is possibly what you currently can find at https://cs230.stanford.edu/files/C1M1.pdf (page 13). It may be roughly interpreted with "low bias learners [in that plot, larger neural networks] tend to benefit from more training examples".

The main reason is that in deep learning the number of training parameters are so many and there is a fact that for each parameter you need at least $5$ to $10$ data to have a good prediction. The reason is a bit complicated to explain but it is related to pack learning and if you insist to know why, I can tell you that in the error term for the test data, you have an overfit term which grows with the number of sample size if your training model is a kind of hypothesis that increases when the number of data increases. In hypothesis with the growth of $O(2^n)$ it is impossible to make the generalisation error same as training error, such as 1NN on the contrary, hypothesis with the growth $O(n^c)$ which are limited to polynomials can have an overfit which can be diminished by increasing the size of training data. Consequently, if you increase the size of your data you can have better generalisation error. Deep learning models obey the second growth manner. The more data you have, the better generalisation you have.