Why does Transfer Learning works better on smaller datasets than on larger ones?
In the article you provide, from page 11 results, I think one cannot conclude that transfer learning works better on smaller datasets than on larger ones.
If you look at the results of transfer learning score values (or RMSE) vs size of learning, it is also getting better while dataset size is increasing (for instance E2 or E5 or E8). So transfer learning does not work better on small datasets.
However, you might be looking at the IMP index which is based on the mean score (or RMSE) of learning with and without transfer learning.

IMP= (1−(WithTransfer)/(NoTransfer))×100

The index is based on two curves.

WithTransfer which will have good performances even at the beginning because when using relevant transfer learning, the model could already extract pertinent information from a very small testing dataset.
NoTransfer which will start with poor performances (difficulty to generalize) and  then increasing with the size of testing data.

The IMP index has then the expected curve you're pointed out, for example with E2 and E5.