This question is far too broad to answer here but the essence of what they follow is based upon the teachings of Rabbi Yoel Teitelbaum.
There are two primary sources that lay out their approach. The first is קונטרס על הגאולה ועל התמורה which is, as best I can tell, not available on the internet.
The second is ויואל משה which can be found here. It is heavily sourced from all parts of the Torah and is something that should be read, if only from the perspective of being able find a way to not hate them. They too are trying to be G-d fearing Jews and to serve HaShem even if according to many, they are misguided.
The essence of the argument is that the final redemption is only supposed to be by G-d directly, unlike all the previous periods of redemption. There is no mitzvah currently to settle the land and that those Jews who have taken possession of the land have done this not to fulfill the Torah, but to deny it. The secular government of the state of Israel is viewed as a government that to a large extent is based upon apikorsus. So all the halachic ideas behind milchemet mitzvah (like a defensive war) for a halachically legitimate government do not apply, particularly with the danger to life by serving in the army. The argument of Bitul Torah only reinforces that position. They also view the effort to establish the state of Israel as an attempt to force the Keitz (the final redemption) which they say was one of the primary causes of the Holocaust.
Opposition to secular Zionism is something that many Orthodox leaders, including the 5th Lubavitcher Rebbe, Rabbi Shalom Dovber, advocated prior to the Holocaust. But this was specifically in regard to the anti-religious aspect of secular Zionism.
From the Alter Rebbe through the fourth Lubavitcher Rebbe, Rabbi Shmuel, there have been many efforts to establish and strengthen the Orthodox settlement of the land of Israel. This was also the position of many of the followers of the Vilna Gaon.
Following the Holocaust, Rabbi Shalom Dovber's son, Rabbi Yosef Yitzchok and his son-in-law, the current Rebbe, took a view that the facts on the ground over-road any opposition from the previous generation. Because so much had been lost, it was essential that Jewish communities be built and strengthened wherever they might be.
There are others, like Rabbi Avraham Yitzchok HaKohen Kook, who compared the secular Zionist settlement movement to the covering of the tachashim used for the Mishkan. It was something needed in the beginning, but then needed to be folded up and put away so that the Mikdash could be built permanently.