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The phrase (as) hungry as a hunter means: extremely hungry, famished, ravenous.

This is one of the similes expressing a comparison with a person or an animal regarded as having a very large or keen appetite—as in, for example: (as) hungry as a wolf, (as) hungry as a horse and (as) hungry as a bear.

A variant of (as) hungry as a hunter, the phrase (as) hungry as a huntsman occurs, for example, in Craving Gravy or Love in the Time of Cannibalism (1997), by the U.S. playwright John Walch—as quoted by Jamie Smith in a review of that play, produced at the University of Texas, Austin, published in the Austin American-Statesman (Austin, Texas, USA) of Tuesday 1^st April 1997 [page E3, column 1]:

Amy L. Washburn’s Gilroy is more likely to listen to the call of her stomach than to think about her former friend sitting on the Chinette plates at their picnic. “I’m as hungry as a huntsman,” she often says.

These are, in chronological order, the earliest occurrences of the phrase (as) hungry as a hunter—and of the earlier form (as) hungry as a huntsman—that I have found:

1-: From The Welsh Embassador, an anonymous play written about 1623—as published in The Dramatic Works of Thomas Dekker (Cambridge (Cambridgeshire, England): University Press, 1961), edited by Fredson Bowers (1905-1991), a U.S. specialist in manuscript editing and bibliography [Volume 4, Act 4, scene 2, page 362]—here, the noun meat has its original sense of food in general, anything used as nourishment, solid food as opposed to drink:

Clowne. This dauncinge ioggs all my dynner out of my belly, I am as hungry as a huntsman; and now I talke of meate, whie does a welsh man loue tosted cheese so well.

2-: From Memoirs of Maximilian de Bethune, Duke of Sully, Prime Minister to Henry the Great (London: Printed for A. Millar, R. and J. Dodsley and W. Shropshire, 1756) [Volume 2, Book 13, page 130, footnote, column 1]:

Le Journal d’Henry IV. relates a little piece of history which I shall set down here. Henry one time hunting on the side of Grosbois, dropt his company, as he frequently did, and came by himself to Creteil, which is a league on the other side of the bridge of Charenton, and that at noon-day, and as hungry as a hunter. Going into an inn, he inquired of the landlady if she had any thing for him to eat [&c.].

3-: From The Artless Lovers. A Novel. In a Series of Letters from Miss Lucy Wheatly in Town, to Miss Annabell Grierson in the Country (London: Printed for J. Wilkie, 1768) [Volume 2, Letter 73, page 288]:

By the time that we return to the Abbey, breakfast is not only ready, but Lord G—m, and Mrs. Farnham are almost ready for it; while Mr. Farnham comes from the boat in which he has been paddling, in his cap and slippers, as hungry as a hunter.

4 & 5-: From Barataria: Or, Sancho Turn’d Governor. A Farce, in two Acts: As it is Performed at the Theatre-Royal, Covent-Garden (London: Printed by J. Almon, 1785), by the Irish playwright and actor Frederick Pilon (1750-1788) [Act 1]—this play also contains the variant phrase as hungry as a judge’s clerk at a long trial:

4-: [page 17]:

Sancho. We have such a craving at our stomach, that we cou’d sit down to dinner with Dapple, and feast upon thistles. Poor fellow! [patting Dapple’s head] he too looks as hungry as a judge’s clerk at a long trial. We may be considered now as the best of all Governors, for we are fasting for our people. Well, if we fast for them now, I will eat most plentiful for myself; odd rat it! we shall never forget I when eating is the word.

5-: [page 19]:

Recorder. You yourself are my Lord—
Sancho. As hungry as a hunter;—therefore, Mr. Recorder, put up your long speech;—and after dinner I’ll put on my night-cap, and hear you go over the whole of it again with composure.

6-: From The Cooper. A musical Entertainment. In two Acts, by the British composer Thomas Arne (1710-1778)—as published in A Collection of the most esteemed Farces and Entertainments performed on the British Stage (Edinburgh: Printed for C. Elliot, 1788) [Volume 6, Act 2, scene 3, page 228]:

Fan. Have you supp’d?
Col. No faith, and am as hungry as a hunter.
Fan. Stay then a minute. (Runs out.
[…]
Enter Fanny with a cake, bottle of wine, and a napkin.
Fan. Come, Colin, we’ll partake of dame Fidget’s present.

7-: From Manner of Hunting the Moose Deer in Cape Breton, published in The Sporting Magazine: Or, Monthly Calendar of the Transactions of the Turf, the Chase, and every other Diversion interesting to the Man of Pleasure, Enterprize, and Spirit (London, England) of October 1792 [page 23, column 2]—the noun punck is a variant of punk, designating soft decayed or rotten wood, as used for tinder:

We all sat down together round the body; and, after asking me how I liked moose-hunting, Benwah bid the boys open the deer the while he collected wood. Then taking his punck-box, he struck a light and made a fire; the boys bringing the heart of the deer and the muzzle, which is the lower part of the mouth, (and a great delicacy with the hunters). We had presently these broiled, and I being hungry as a hunter, made a most comfortable repast.