Why do speakers with different dialects feel that some forms of language are prejudice towards them?

To answer this question, we need to look at the history of languages in Europe and other continents.
In each of the nation-states of modern Europe, there were and are multiple dialects spoken of that country's language. At an earlier time, all the dialects of a given language were on a more-or-less equal footing, and in most cases, it was only in the late middle ages or even more recently that a particular dialect gained special prestige and became the standard (especially literary) form of the language. For instance, in Italy it was Tuscan, the dialect of Florence, which from Dante's time on became "standard" Italian—though there were (and still are) other dialects spoken in different parts of Italy, some of which are different enough from Tuscan that they may not be recognizable to the uninitiated as part of the same language. In England, the dialect of London (unsurprisingly) became the standard from Chaucer's time in the late fourteenth century onwards. In Germany, it was the dialect of Saxony that became standard German from the time Martin Luther translated the New Testament into German in the sixteenth century.
In every country, people learn the standard version of their language in school, though a dialect may be spoken in their home and in their region overall. To outsiders, a given dialect, because it doesn't have the prestige of the standard version, is often (wrongly) looked upon as if it were illiterate speech. But originally there were no standard versions, so one can see why this attitude would be resented by people who do speak a different dialect.
One could argue that it was only by a kind of accident that Tuscan, for example, became standard Italian. There is no objective reason that Sicilian or Neapolitan should not have a claim to the same validity or integrity as a language. It took centuries for the European countries as they exist today to become unified, and everyone tends to have a dual loyalty: to the nation as a whole, and to the particular province or region where they were born. In Germany, for instance, many people from Bavaria see themselves as Bavarians first and Germans second. So if one's regional form of the language is marginalized or dismissed by outsiders—in other words, when a value judgment is implied in outsiders's view of it—it's understandable that those who speak it would see that attitude as high-handed and possibly demeaning.
In the US, the same situation exists but in a less striking way, because the dialects of English spoken here have less variation among them than the dialects in Europe. There are different reasons for this. Even from before independence, the new Americans came not just from England but from other countries. All had to learn the standard form of English in order for communication to be universal among them. With the major and obvious exception of the Native Americans, people in the US did not have the kind of centuries-old roots in a local region that existed in the Old World. Nevertheless, as we can see, there are obvious differences in speech throughout the country, and it's unfortunate if some people still tend to make value judgments with regard to those differences.