The relevant history goes back centuries, but, in a nutshell, the current intractable conflict exists mainly because in 1967 Israel conquered and occupied what are now known as the Occupied Territories, the West Bank and the Gaza Strip, which were and remain full of Arabic-speaking Muslim Palestinians. (The West Bank was Jordanian territory at the time, but Jordan wants no part of it any more.) It must have seemed a defensive necessity at the time; the Jordan River is a more defensible frontier.
Israel conquered these territories but, except for East Jerusalem, did not annex them; so these territories remain neither in Israel nor out of it, and the Palestinians are citizens of no country; they are, like a colonial population in the old days, or the blacks in South Africa under Apartheid, in effect ruled by Israel without having any voting voice in it. And they don’t like that. And it really fucks up their economy, too. The Israelis nabbed all the best-watered land, and the Palestinians can’t even travel to or trade with Jordan. There seems to have been the idea current in 1967 that the Israelis could make those territories truly Israeli by colonizing them with Jews, but, there just aren’t enough settlers compared to the Palestinians. See this map – everything in dark pink is an Israeli settlement, everything in light pink is a restricted military area. Note that everything near the river is pink.
There are 3.7 million Palestinians in the West Bank and Gaza, and 1.3 million in Israel proper. Most of the latter are Israeli citizens. There are 5.9 million Israeli Jews, including 300,000 in West Bank settlements. So, all told, they outnumber the Palestinians – but not by much.
So, annexing the Territories and makng voting Israeli citizens of the Palestinians . . . would not be the end of the “Jewish State,” not immediately, but Israelis fear the high birthrate among Palestinians, and the “Law of Return” the latter want enacted that would allow Palestinian refugees (or their children or grandchildren) from all over the world to move to Israel/Palestine . . . those factors might eventually lead to the Jews being outnumbered in what was founded to be their own country. So, nobody in Israeli politics seriously considers a “one-state solution.”
OTOH, a “two-state solution,” with the OTs forming an independent Palestinian state, also frightens the Israelis, because they’re not confident the Palestinians would settle for that. They fear the result would be yet another hostile neighbor on their borders; and besides, the 300,000 Israeli settlers now in the West Bank would have to be either uprooted and moved west, or left to the mercies of the new Palestinian government, both politically unsavory options. And they’re not at all encouraged by the way things have gone in the Gaza Strip since the Israelis pulled out of there. Anti-Israeli forces have been launching rocket attacks from Gaza into Israel. In response, the Israelis, rightly or wrongly, have judged it necessary to impose an embargo so that the fractious Gazans cannot trade with the world or even with their only non-Israeli neighbor, Egypt.
And that’s why this conflict goes on and on and on. Because there is no obvious solution that is more-or-less acceptable to all parties.
And, of course, in the above analysis, “all parties” means only those people actually living between the Jordan and the sea. It does not simplify things that most Muslim countries have an ideological hostility to Israel, and the U.S. and many other Western countries support Israel.