The suggestion by @MCW in comments -- that veterinarians check the teeth of livestock -- is unfortunately probably the right track to follow. Southern Slave traders and slave buyers routinely stated that they looked at teeth to try to judge health, age, and vigor. During the antebellum period, people were often driven to slave market centers from hundreds of miles away, and buyers had relatively little reliable information about where they had come from, who they were, their life history, their physical abilities and health, etc., which they would try to judge by "inspecting" them. The assumption was that a slave trader might be ignorant, fudging or lying about these things to make a sale. Buyers apparently did often pose at least some questions directly to the enslaved people themselves; but didn't put much trust in what they said, partly because of the overwhelming arrogance of privilege, racism, etc., and also partly because the person they were asking was, after all, physically at the mercy of the slave trader and might just have to say what they were told to.
Walter Johnson has a useful discussion of this in Chapter 5 of Soul by Soul (2009), which deals with the daily practices within slave markets in New Orleans, one of the largest antebellum slave-trading centers for the Deep South. Chapter 5 deals closely with the ways that white men buying slaves in the New Orleans markets (for themselves or as agents for another) would examine people and their bodies to try to judge "likely" hands. Buyers certainly did examine hands and arms, muscles, and physical build (141ff, et seq.); they also looked for signs of a person's past history (for example injuries from accidents or scars from whipping). And as you noticed, they also pawed at people's mouths to check teeth and gums.
Johnson thinks (drawing on contemporary observers like traveler's literature) that they most likely got the practice from horse dealing and veterinary practice. Here's what he says in the book; endnotes provide more information about Johnson's references. The references to court cases are from the docket record of disputed slave sales (in which a buyer demanded a refund from the slave trader) that reached the Louisiana state Supreme Court (Johnson 2009, 12). The part about "the whiteness of sick slaves" refers back to some previous passages where Johnson describes how pallid color in a black body was sometimes taken to be a sign of weak health or degenerative disease. Boldface emphasis is mine, not in the original text. (It may go without saying, but much of what's described in this chapter is extremely ugly and can be hard to read.)
Chapter Five. Reading Bodies and Marking Race.
[...] Asked to explain what they looked for in a slave, most slave buyers would have responded with the word "likely." Today the word means probable, but as slave buyers used it was as much description as it was a prediction. As they singled out the "likely" from among the many they saw in the pens, slave buyers made detailed inspections of people's bodies which went well beyond the traders' advertisements and the age, sex, and racial designation that were commonly recorded on an Act of Sale. The standard slave inspection, as one buyer described it, went like this: "my inspection was made in the usual manner: their coats being taken off and the breast, arms, teeth, and general form and appearance looked at." The whole process, according to another buyer, might take anywhere from fifteen minutes to half an hour, and bargaining might be stretched over three or four days. The inspections, at least at the outset, were public. The white male buyers in the yard mingled as they walked the rows of slaves; they observed the inspections made by one another and shared their own reckonings of particular slaves; they talked about and joked the slaves standing before them.9 [...] [138ff]
[Endnotes, Ch. 5, n. 9.] Lyons v. Kenner and Taylor, #4770, Z Rob. 50 (1842), testimony of S. Bennett; UNO; White v. Slatter, #943, 5 La. Ann. 27 (1849), testimony of S. F. Slatter, UNO; Huntington v. Brown, #15, 7 La. Ann. 48 (1865), testimony of D. Donvan, UNO.
[...] In a gesture that many observers connected with the practice of the horse trade, buyers thumbed their way into slaves' mouths to look at their gums and teeth. The whiteness of sick slaves first appeared in their mouths, according to Southern medical science. Samuel Cartwright saw the signs of "Negro Consumption" in the whiteness "of the mucous surfaces lining the gums and the inside of the mouth, lips and cheeks: so white are the mucous surfaces that some overseers call it the paper-gum disease." Cartwright's description suggests that he thought it routine for a slaveholder to pull back slaves' cheeks and finger their lips. And indeed slave trader A. J. McElveen referred to teeth again and again in the letters he wrote to his boss: "Very badly whipped but good teeth"; "Likely except bad teeth"; "very likely, has good sense, fine teeth."19 McElveen may have been choosing slaves in the same way he would have chosen a horse: judging their age by the condition of their teeth. Or, more simply, McElveen may have been looking at teeth and gums because he knew that his buyers would. The rote practice of the market could produce its own standards of comparison: teeth did not need to be a practical sign of anything in particular to be compared to one another. [...] [142]
[Endnotes, Ch. 5, n. 19.] Marin v. Michel, unreported Louisiana Supreme Court case #5298 (1858), testimony of Jacob Capan, Louis Caretta, and David Keane, UNO; Fredrika Bremer, Homes of the New World, 2 vols., Mary Howitt, trans. (New York: Harper Brothers, 1853), II, 203; Charles Richard Weld, A Vacation Tour in the United States and Canada (London: Longman, Brown, Green, and Longman, 1855), 300; James Redpath, The Roving Editor, or Talks with Slaves in the Southern States (New York: A. B. Burdick, 1859), 249-250; Henry Bibb, Narrative of the Life and Adventures of Henry Bibb, an American Slave, Written by Himself (1845), in Gilbert Osofsky, ed., Puttin' on Ole Massa (New York: Harper & Row, 1969), 114; Peter Randolph, Sketches of Slave Life: Or Illustrations of the Peculiar Institution (Boston: Published for the Author, 1855), 52; A. J. McElveen to Z. B. Oakes, July 10, 1853
, July 29, 1852, May 10, 1854, in Drago, ed., Broke by the War, 43-51, 80; Cartwright, "Diseases and Physical Peculiarities of the Negro," 212.
—Walter Johnson (2009), Soul by Soul: Inside the Antebellum Slave Market, 138ff, 142. Boldface added.
In Southern agrarian society, a buyer would look at horses' teeth and gums to try to judge age (from wear), nutrition and health (with bad teeth or discolored or inflamed gums indicating possible malnutrition or chronic disease). Or, as Johnson notes, doing what you do conventionally to inspect and make a judgement could also have basically ritual functions, for white men to go through the motions that are expected to demonstrate due diligence and expertise, even apart from any direct practical value that it might have. In any case, it seems that the practice was transferred; as with many things in Southern slave society, white people who dealt in enslaved black people more or less consciously adapted practices that were familiar from dealing in livestock to the ways they judged and treated enslaved black people.
