It's a big mixed bag, but a few key reasons are:

• The IDF represent an armed faction of the secular Jews of Israel. They will make a distinction of Israelis and Israelites, with the Orthodox being the latter. Before the Romans, there were the states of Israel (Tel Aviv) and Judea (Jerusalem), so that separation continues in their mind.
• If their kids were conscripted, they fear the secular influences upon them and it is a driving factor in their opposition to conscription.
• Surprisingly, there are factions of Orthodox who don't believe the state of Israel should exist, at least until the Messiah returns. So they oppose the armed faction of the illegitimate state.

There is so much I can't even begin to explain, and I'm sure some Israelis would even have corrections from what I learned over there. The Orthodox live a decent lifestyle which is 100% paid for by the Secular community, so they can spend their days studying and causing trouble for others. When you hear about armed groups shooting at Palestinians and burning down homes, you're making a safe bet if you assume they're Orthodox.