In Israel most people are Jewish (by nationality). And this thing doesn't quite work out the way a person living in a Christian protestant country would expect.

Like most (all?) countries, you have to make certain statements with appropriate ceremony (not to mention pomp and circumstance). For instance, at the end of Basic Training in your military service, you swear allegiance to your country etc.; Knesset members swear compliance with the laws of the state.

I'm a secular Jew, which means I'm emphatically not religious. So I have no problems swearing, taking an oath, or whatever (God is not invoked in these matters!).

But a religious Jew cannot swear or take an oath, even if God is not involved. The thing is that He is always potentially involved (even if not explicitly mentioned). And you're not allowed to take His name in vain. Almost any oath could be in vain as far as He is concerned (Maybe He doesn't care about Knesset laws? Or maybe He would want you to be a pacificist? You just don't know!). So religious people don't swear or take oaths (and on Yom Kippur a special ceremony -- Kol Nidrei -- is held to free one of oaths mistakenly taken!).

Instead, religious people declare the appropriate statement; non-religious people swear it. The choice is open.