For nearly 2,000 years,
Jews have been, for the most part, unable to live in the
land of
Israel. During this time period, known to most Jews as the Edomite Exile (based on the fact that the rulers that ejected them from the land were a part of the
Roman, or Edomite empire,) different attitudes have been held about return to the land of
Israel
In the Talmud, Tractate Kesubos, 110b, there is a discussion of Mishnaic teachings saying that a person who lives outside of the land of Israel (that is able to move), it is as if they worshipped idols. Then there is a discussion based on scriptural sources about whether Jews are still obligated to observe an oath that was made concerning whether Jews could return to Israel before the coming of the Messiah. Based on this peice of the Talmud (and several other sources), a number of authorities in the last dozen centuries discuss whether this law is still in effect. The discussion became more and more heated as the land of Israel was settled in the early 1900's. Many prominent Rabbis were of the opinion that, while any individual could certainly move to the Land of Israel, the nation as a whole was not to return until the Messiah came and established a nation with the guidance of prophets, and rebuilt the temple in Jerusalem.
As the modern State of Israel was founded, the vast majority of religious Jews still beleived that it was not acceptable to have a state of Israel, but a minority of religious jews decided to go to Israel even though almost no leaders of the generation said that the state was acceptable in Jewish law. Various different ideologies motivated the Aliyah (ascent) of Jews to Israel, including opposite ends of the spectrum. "Modern Orthodox" Jews had a philosophy combining many of the tenets of the previously stricly secular zionist movement, and the ideals of religious, (but mostly American) Judaism . The Hassidim and other European Jewish communities moved for purely religious reasons, not to support the state, and were mostly of the opinion that while it was not acceptable to have a 'Jewish State', they would live there anyways.
After the Six Day War, many religious Jews, in America, Israel, and elsewhere felt that the amazing victory was a sign that G-d had decided to intervene on behalf of Israel, and clearly if G-d helped the country to repel such incredible odds, it must be what was correct. A minority, however, still felt that it was absolutely forbidden to have a state of Israel. Within this minority, there was a radical splinter group, The Neturei Karta, that for the most part, has dissasociated itself from all other religious jews (including most of those who feel that the state of Israel is wrong.)
Neturei Karta has staged a number of protests, both in New York and in Israel. Many of these protests occur on Saturday, which is the Jewish Sabbath. In several cases, leaders of the movement have performed acts that clearly violate the laws of Sabbath, and have upset a sizeable portion of people formerly associated with their views. In addition to this, the protests are often held with Palestinians who are known to have sponsored, or been involved with personally, acts of terror agaist jews, which is also seen as a major point against them. Even if people were willing to discuss the points they raise, the involvement in acts that put them well outside the realm of Halacha, estranging many possible followers.
In short, the arguement that Neturei Karta puts forwards is well grounded in terms of purely Halacha reality, but in view of the acts that they have done, and the fact that they are seen but almost everyone as a radical group completely destroys their credibility with Jews around the world.