This is from AA Milne's The House at Pooh Corner (a humorous story book for children). The characters in the Winnie-the-Pooh stories were named after Milne's son, Christopher Robin Milne, and his collection of animal toys.
'How do you do?' is a conventional middle and upper class greeting in England, possibly more common in 1928, when the book was written, than now. It is usually said to someone one has not met before. The correct response to 'How do you do?' is to repeat it back. Thus Eeyore should really, in the adult world, have said 'How do you do?' back to 'everybody'. There is humour here because 'everybody' (the other 'toys') already know Eeyore and so the greeting is misplaced, but the toys are, in Christopher-Robin-world, somewhat ignorant of adult conventions.
Eeyore, as one might expect, responds inappropriately, as if 'How do you do?' is a real question. He further misinterprets it by responding as if it was 'How much do you do?', and says something idiomatic that means 'Not very much, really'. One may deduce that Eeyore says something like 'I don't, not to notice'. We can emphasise that something is not very large (e.g. the extent of what Eeyore 'does') by adding 'not that you'd notice'. Example: Mother: Has Michael done any school homework today? Father: Not that you'd notice. Eeyore is, as Kate Bunting notes, 'famously negative'.
Books for children are often written to be rewarding for a child of the target age group, by showing characters whose understanding of the world is somewhat less than the child's. Thus the child is amused and made to feel knowledgeable.
The humour, which might be a little cruel if applied to humans, is that 'Eeyore is stupid'. It is worth noting in this regard that the use of 'How do you do?' and of the 'correct' response, was (and probably still is) a 'class marker' in England.