Based on the description you’ve posted it seems the authors call the output features embeddings.
This might be a bit confusing, as there are nn.Embedding layers, which are apparenrently not meant here.
If I understand the use case correctly, you could store each output feature of the VGGish model with its corresponding target, create a new classification model, and use these output features + targets to train this new classifier.
According to their claim, this classifier can be “shallower”, as the “embeddings” are so great. 
Let me know, if that makes sense.