This question arises in the context of the traditional philosophical enquiry into the nature of knowledge. We might summarise that enquiry as: under what conditions is it correct to say that a person X knows that a proposition P is true? To emphasise the point: the analysis is not concerned with the conditions under which P is true or accurate, but the conditions under which it is correct to say that X knows that P. A fairly standard approach is to say that X knows that P if (a) X believes P, (b) P is true, and (c) some additional constraint is present that connects the belief with the truth of P. Some plausible candidates for (c) are that the belief was obtained in a reliable way, or that is was caused in an appropriate way, or that the belief counterfactually tracks the truth of P, or that the truth of P was not a matter of luck.
The problem is that there is no simple answer to the question of what knowledge is. In real world situations, whether a person can justifiably claim to know that P depends on such factors as who is speaking, on what occasion, in front of what audience, what common knowledge there is between themselves and their audience, and what is at stake. These are said to be pragmatic factors. The term pragmatic encroachment was introduced in 2004, though arguably the idea behind it goes back to 2001.
Consider some examples. Suppose Alice is taking her young son Bob to the zoo. She points to a zebra and tells Bob this is a zebra. By doing so, she is claiming to know that this animal is a zebra. Bob might even ask whether she knows this is a zebra, and she will justifiably say: yes. Alice, like most of us, is competent at identifying zebras. Now suppose Charlie, who is an expert zoologist, asks Alice: How do you know this isn't a horse painted to look like a zebra? Alice does not have the expertise to tell the difference. She may judge it unlikely that a zoo would display a painted horse, but she can hardly claim to be certain that it would not. When it was just a question of telling her son that this is a zebra, such considerations were irrelevant, but now that the question of the animal possibly being a painted horse has become salient, it makes a difference. Alice cannot justifiably claim to Charlie that she knows this animal is a zebra. The pragmatics of the situation, i.e. whom Alice is talking to, together with her and their level of expertise, makes a difference as to whether she can claim to know the animal is a zebra.
Suppose we ask Carol what blood group she is. She replies: 'A'. She believes this because she had a blood test a few years ago, and this is what is written down on her blood donor card. And it is true. For most purposes it would seem reasonable to say that Carol knows her blood group. But if she is about to receive a blood transfusion and a mistake will cost her her life, she might well say she does now know, or at least not for certain. A confirming test would be prudent. Here, the issue of what is at stake becomes material to the question of whether Carol knows her blood group.
Examples like these arise because there is a tension between the requirement on the one hand that knowledge should be reliable, or truth-tracking, or caused in an appropriate way, or free from luck, and the fact on the other hand that we cannot legitimately claim absolute certainty about anything. To require certainty would rule out knowledge altogether, while to have a single fixed constraint that is a sufficient condition allows for counterexamples. Pragmatic encroachment is the position that what counts as a sufficient condition for knowledge depends on the speaker, the audience, the circumstances, the common knowledge, what is at stake, etc.
Pragmatic encroachment is not unique to the analysis of knowledge. Pragmatics is also concerned with what utterances mean and under what conditions they may be true. Sentences do not typically have a single determinate meaning or truth value that is independent of the pragmatic circumstances in which they are uttered. As to being specific, the extent of the effect of pragmatics is not readily measurable in a quantitative way. One can try to quantify degrees of uncertainty - Bayesians do - but one is still left with the fact that there is no particular single value epsilon, such that one can claim to know a proposition A if the probability of A is greater than 1 minus epsilon. The appropriate value of epsilon would itself be subject to pragmatic factors.