Every adjustment the model made in the learning loop was driven by the training examples. Those examples are the only evidence the model has ever been corrected on. So a low loss means the model fits that particular set of examples well — it does not yet tell you whether the model captured anything that carries over to a new email, a new photo, or a new customer. The training set is like the practice questions you have already seen the answers to. Doing well on them is encouraging, but it is not the same as doing well on the exam.
The purpose of building a model is to use it on data that did not exist when it was trained. If it only works on the training examples, it has not learned anything useful — it has just recorded answers. Performance on unseen data is therefore the only test that matters for real use.
Suppose a spam filter is trained on 1,000 emails and achieves a loss near zero — it labels almost every one correctly. Then a new email arrives, written in a style the filter never saw. If the filter still classifies it correctly, it learned something about spam. If it fails, the low training loss was misleading. The filter's real job begins with emails it was not trained on.