When you fillet a fish, for example, you remove the bones while keeping the outer structure the same. When you look at a whole fish in fishmonger's shop, you may not be able to tell that the bones, and also the guts, have been removed. This image is used to describe what is happening to the man's brain as the tumour destroys sections of its interior. It appears that different areas of vocabulary are stored in separate 'compartments' of the brain. So as the tumour progresses, the man loses the ability to retrieve the words stored in the damaged areas of the brain while he can still retrieve those in the areas that are intact.

To 'fillet' a fish is to cut it open and pull out its guts. You can search 'filleting a fish' on youtube (it will be gross). To 'fillet' anything else means to cut out lots of parts of it. This person's vocabulary (words that they know) has lost a lot of things (been filleted). As a side note, 'fillet' and 'filleted' can be pronounced in different ways depending on where you are. In informal English and in lower social classes, we'd say 'fill-it' and 'fill-it-id'. In formal English or in higher classes, 'fill-ay' and 'fill-ay-ed'.