The word "biting" in relation to Swift means harsh, cruel, or cutting. There's little doubt that this is an accurate description of his writing. Compared with the more sedate, gently mocking satire of his contemporary Alexander Pope, Swift's is much more shocking and outrageous. We can consider one example: A Modest Proposal. In the elevated style and tone of a learned scientific research paper, Swift puts forward a suggestion for solving many of Ireland's most pressing economic problems—getting the poor to sell their children for meat.
And in Gulliver's Travels, Swift is especially scathing of British politics and its personalities and institutions. The political conventions of Lilliput, for example, bear more than a passing resemblance to those existing in the Britain of Swift's day. At the Lilliputian court, whenever a vacancy arises, those interested in serving the Emperor are required to participate in a degrading rope dance ritual for His Excellency's entertainment and delectation. Whoever can jump the highest without falling over gets the job. Swift is making it perfectly clear that in Lilliput, as in Britain, achieving high office has nothing to do with brains, talent, or ability; it's all about doing the reigning monarch's bidding.
What makes Swift's satire so cutting is his rampant misanthropy. Throughout Swift's writing, he conveys the impression that he really doesn't like people all that much. This puts a large distance between himself and the objects of his satire, providing him with a more disinterested perspective, as well as allowing him to be as vicious and as cruel as he likes. However unpleasant this approach may come over at times, there's no doubt that it's highly effective, and amusing, in its own unique way.