In Letter 144 (p.174) he states:
As a story, I think it is good that there should be a lot of things unexplained (especially if an explanation actually exists)....And even in a mythical Age there must be some enigmas, as there always are. Tom Bombadil is one (intentionally).Not a very satisfying answer for curious hobbitses!
As for what Tom actually is, in Letter 19 (p.26) Tom is "the spirit of the (vanishing) Oxford and Berkshire countryside".
In Letter 144 (pp. 178-179) he writes that "Tom Bombadil is not an important person - to the narrative." And "he represents something that I feel important." Why was he left in LOTR?
I would not, however, have left him in, if he did not have some kind of function. I might put it this way. The story is cast in terms of a good side, and a bad side, beauty against ruthless ugliness, tyranny against kingship, moderated freedom with consent against compulsion that has long lost any object save mere power,and so on; but both sides in some degree, conservative or destructive, want a measure of control. But if you have, as it were taken 'a vow of poverty', renounced control, and take your delight in things for themselves without reference to yourself, watching, observing, and to some extent knowing, then the question of the rights and wrongs of power and control might become utterly meaningless to you, and the means of power quite valueless.And later on in the same paragraph: "Ultimately only the victory of the West will allow Bombadil to continue, or even to survive. Nothing would be left for him in the world of Sauron."
Restated in Letter 153 (p. 192):
In historical fact I put him in because I had already 'invented' him independently (he first appeared in the Oxford Magazine) and wanted an 'adventure' on the way. But I kept him in, and as he was, because he represents certain things otherwise left out. I do not mean him to be an allegory - or I should not have given him so particular, individual, and ridiculous a name - but 'allegory' is the only mode of exhibiting certain functions: he is then an 'allegory', or an exemplar, a particular embodying of pure (real) natural science: the spirit that desires knowledge of other things, their history and nature, because they are 'other' and wholly independent of the enquiring mind, a spirit coeval with the rational mind, and entirely unconcerned with 'doing' anything with the knowledge: Zoology and Biology not Cattle-breeding or Agriculture.Hope this helps out some! Since I was driving myself crazy trying to figure out who/what Tom is, I thought I'd have better luck finding out what Tolkien intended him to be.