I'll be honest I haven't a clue when it comes to politics.
As I understand it, regarding conscription for ultra-orthodox, it's not that fighting itself is considered a sin (which would be closer to what a conscientious objector might believe), it's more complex and multi-layered than that. Ultra-orthodox Jews strongly regard the studying of the Torah as just as essential for the security of the country as fighting. (So do religious Zionists, who often combine IDF service with Yeshiva study, in the Hesder program). This has biblical roots as well, where some historical commentators note that when the Israelites went out to battle (don't ask me the source sorry!), the army was split into three parts; 1 part combat, 1 part Torah learners, and the third something else (support?). The idea of Torah learning being essential for defence means many Charedim believe that drafting those who are engaged in Torah learning actually removes a layer of defence from the country, and leaves us far more vulnerable, and that those engaged in Torah learning are fighting on the spiritual front for our protection.
(Some religious zionists counter argue that this was a milchemet mitzva (a war that is a mitzva, a commandment) and therefore everybody is obligated to fight. But there's debate around whether it can constitute a milchemet mitzva or not. )
Another layer is that the ultra-orthodox community is very separate, and deliberately pursues a very closed and sheltered way of life. The IDF as an environment exposes these very sheltered young men (and women) to ways of life that is seen as antithetical to these communities way of life, and they are worried that this will cause their children to leave their way of life and communities. The fact that the army is mixed gender, for example, is highly problematic. Concerns around facilitation of religious observance and kashrus are also part of the issue. The establishment of Charedi Chashmonaim brigade was an attempt to resolve this issue, and work on Charedi army integration. However, some regard forced conscription as an attempt at forced assimilation and secular integration - an attack on their way of life, which also taps into historical tensions as well, between Charedi communities and the state.
Some Charedi communities in Israel pre-date the state (e.g. the Old Yishuv), and were opposed to the rise of political zionism. A small number of ultra-orthodox sects are opposed to the state of Israel altogether, and believe we should be waiting for Mashiach, before re-establishing a Jewish state in the land. Neturei karta (who are frequently quoted by antisemites) are one such group who believe this - they were a breakaway group who originated in the old Yishuv. However, NK are pretty much condemned across the board nowadays.
In short, it is political, historical, cultural, and religious.
(Disclaimer: This is just my understanding of the issues)