There are two very different ways a model can achieve a low training loss. It can learn the pattern that actually connects inputs to outputs — the features that make an email spam, the visual cues that make an image a cat. Or it can simply record which training inputs had which answers, without extracting any pattern. The first is generalizing; the second is memorizing. Both produce low training loss, but only the first survives contact with new data.
A student who memorizes the answers to last year's exam questions can score perfectly on that paper. Give them a new paper with different questions and the memorized answers are useless. A student who understood the underlying principles can handle questions they have never seen. The model that memorizes is the first student; the model that generalizes is the second. The training loss cannot tell them apart — only new questions can.
Memorizing is not a rare failure mode — it is often the easiest way to reduce training loss, especially when the model has many adjustable settings relative to the number of examples. With enough settings, a model can fit almost any set of training answers exactly, including the random noise in them, without learning anything that transfers.