This is an exceedingly complicated question due to all of the politics involved, but the simplest way to answer is this: because it’s where the events depicted in the Torah took place, and the land that the Jews have collectively commemorated as the homeland from which they were expelled by the Romans in the 1st century. (The question of Judaism’s attitude toward Christianity and Islam isn’t really relevant in this.)

The idea of establishing a Jewish homeland came out of late 19th century Europe—this is a time when ethnonationalism (the idea that ethnic groups—nations—should have their own political entities with clearly defined borders—states—as their exclusive ‘homeland’) was on the rise in Europe. Germany had unified, Italy had unified, various Slavic entities and Greece were splitting off from the Ottoman Empire, there were Hungarian and Czech nationalist movements sprouting in the Austro-Hungarian empire, and there was concern among Jewish intellectuals in Europe that anti-Semitism was on the rise because Jews had traditionally been seen as different and alien (note: I am vastly oversimplifying here).

Prominent Jewish thinkers were troubled by episodes like the Dreyfus Affair in France, in which a high ranking army officer was convicted of treason on very flimsy evidence, largely because his real offense was being a high ranking Jewish officer in the French army. And the idea had been circulated that the Jews, also, needed a homeland because, as a minority population, they could never be certain that the majority in any given country would protect them (as opposed to turning on them—in Europe, in particular, there was a long history of this).

Theodor Herzl, who had proposed the idea of a Jewish state (this movement became known as Zionism), initially suggested that the state should be acquired through the purchase of land. Palestine was always the most attractive option because of its intrinsic value in Jewish culture, although other ideas were circulated, including a plot of land in Patagonia, and the island of Madagascar. A group of Zionists met with Ottoman officials (modern day Israel was part of the Ottoman Empire at the time) hoping to convince the sultan to let them buy the land. The Ottomans—a large, multi-national empire—were faced with their own issues of nationalist movements that had caused the empire to rapidly lose territory over the course of the 19th century, and—realizing that the idea was that this area would eventually become independent from the Empire—responded instead that they welcomed Jewish immigration from Europe to anywhere in the Empire except for Palestine.

During the early 20th century before WWI, there was a lot of “stealth” migration, in which Jews officially told Ottoman authorities that they were moving to other parts of the realm (usually Salonica, which is now in Greece, or Smyrna—now the city of Izmir in Turkey; both had large Jewish populations descended from the Jews expelled from Spain in the 15th century), and then quietly moved to Palestine some time thereafter. (The Ottomans weren’t that great at controlling internal movement.)

WWI changed the political landscape, and really was the point at which the final decision about where the Jewish homeland would be was made (one could argue that Palestine was, from the beginning, the only real option on the table, but the end of the war was when it became irreversible). The Ottoman Empire found itself on the opposite side from the British and French. The British were encouraging the Arab population under Ottoman rule to rebel against the Ottoman (Turkish) leadership, offering them British support for independence in return. The British Zionist Agency managed to convince the British Foreign Secretary, Arthur Balfour, to get approval for an official statement that the British government supported the creation of the Jewish homeland in Palestine (the infamous ‘Balfour Declaration’ of 1917).

This is, unfortunately, the same time that everything got very messy. The British and the French worked out a deal for a post war Middle East that would be under their supervision (this is the Sykes-Picot Agreement, conducted in secret). The British had also been leading along the emir of Mecca, Husayn bin Ali (great grandfather of the current king of Jordan), promising him rule over the Arab territories that had been under Ottoman rule in exchange for helping to defeat the Empire. This was openly known, and, in fact, Chaim Weizmann, the head of the world Zionist organization and later president of Israel, had actually been in touch with Prince Faisal, Husayn’s son. They had a lengthy correspondence discussing the creation of an autonomous Jewish state in Palestine that would exist within the new Arab kingdom; negotiations were going fairly well when the Sykes-Picot agreement became publicly known.

Once that happened and it was revealed that the Arabs weren’t going to get their promised independence, but rather European controlled mandates would be imposed, everything was set. The Arabs still consider this Britain’s great betrayal of the Arab cause. Britain established a mandate over Palestine, Transjordan, and Iraq, and the rest of history is well known.

/u/khowega's answer addresses the political motivations behind Zionism, so I want to briefly touch on the significance of that region. Like they said, Israel has religious and historical significance to the Jews. Judaism isn't just a religion, but an ethnicity as well, with deep ties to the land of Israel (and those ties helped influence Israel's importance to Islam and Christianity, as you noted.) According to the Tanakh (the Hebrew Bible, featuring the Torah as well as the Books of Prophets and Books of Writings), the land of Israel is the Promised Land that God had promised to Abraham when He made a covenant to make his descendants a great nation; Abraham's grandson Jacob was renamed Israel (meaning "wrestles with God") by an angel, and the descendants of his 12 children became the Israelites who Moses leads to the Promised Land, which they named Israel. I can't comment too much on the historicity of the biblical narrative, but there's a lot of legend mix in with certain inklings of fact—I'll point you to this answer from /u/luutarhur1 for more on that topic.

Regardless of the veracity of the Bible, an ethnoreligious tribe of people who are or become called the Israelites do in fact wind up living in Israel, believing in the story of the Torah and following its religious and legal laws. They build the Temple, they circumcise their sons, they follow kosher, they keep the sabbath… all those Commandments that Jews throughout history continue[d] to observe (well, not all… some traditions got modified or abandoned over the millennia, depending on various circumstances). Various political powers like the Persians and the Seleucids conquered the region and Jews became part of their dominion, and sometimes they were able to practice freely and sometimes they were forbidden and/or exiled. King Antiochus IV of the Seleucids ransacks the Holy Temple in the 160s BCE, leading to the Maccabean Revolt, where Jews from Modin were able to kick out Antiochus, rededicating the Temple (which is what Hanukkah celebrates, the name meaning dedication) and installing the Hasmonean dynasty. This lasts until about 63 BCE when Rome conquers the land of Israel. Jews were spread out across the Roman empire and beyond, and were therefore not confined to just Israel, but Israel and Jerusalem always were places of religious and historical importance to the Jews, regardless of where they were.

There were Jews who protested and fought for independence under Roman rule, like the revolt in 60s CE and the mass suicide at Masada in 73 CE, but these efforts led to the destruction of Jerusalem and the temple, as well as the destruction of many Jewish villages. Bar Kokhba led a rebellion in 135 after Hadrian tried to Hellenize the area Jerusalem was in, and the consequence of this failure led to further repression for quite a while, such as forbidden entry to the land of Jerusalem. Through a series of exiles and migrations, most of the Jews left Israel, spreading across the Middle East into North Africa, across Europe, and throughout Asia, and the impacts of this Diaspora naturally affected them for millennia: pretty much everywhere they went, Jews faced some sort of persecution, and have been forced out of many many places at a moment's notice. But throughout the Diaspora Israel remained a holy location with religious significance, and there was always hope that someday Jerusalem and the Temple would be restored, and since the 15th century Jews would pray on Passover and Yom Kippur L'Shana Haba'ah, "Next year in Jerusalem."

So, the land of Israel was chosen as a return to the religious and historical homeland of the ancestors of contemporaneous Jews. Though, as /u/khowaga notes in their summary about early Zionism, Israel was not the only land considered for a Jewish homeland—the primary motivation for Zionism wasn't return to the Promised Land, just a place to be safe when they faced discrimination more or less everywhere in the world. But the land of Israel had deep significance to Jews worldwide, so it was chosen.

Further Reading

Brenner, Michael. A Short History of the Jews. Princeton University Press, 2010.

I know it's been a month, but neither of the two questions mentions the religious significance of the land besides it being a setting for the Old Testament's stories. What's left out there is that many of the land's sacred and holy sites, including those holy to Muslims and Christians, are indeed sacred to Jews (and were first, obviously).

Consider the following:

• Where is the Tomb of the Patriarchs, the burial site purchased by Abraham for himself and his wife Sarah, Isaac and Rebekah (Isaac being Abraham's son), and Jacob and Leah (Jacob is Abraham's grandson)? In Hebron, an area the Israeli government finds quite important and which was part of a motivation for the attempt to create Israel in the entire British Mandate (which would've included Hebron, in the West Bank), before Jews accepted the UN Partition Proposal (which would've left it in Arab hands nonetheless).

• Where is the Western/Wailing Wall, the outer retaining wall of the Second Temple (the Temple being the holiest site in Judaism, though many confuse it with the wall), located? In Jerusalem, of course. The wall is the only part of that holy site still standing, according to Jewish theology and perhaps some archaeology (though that is contentious), and Jews regularly pray there and leave notes in its cracks with prayers that Jews believe go straight to the ears of God. You may remember the wall's significance for another reason: in Islam, Muhammad's journey to ascend to heaven in Jerusalem on the back of a winged steed led him to the Temple Mount, but he tied the steed near (or to) the wall.

• Where is the Temple Mount located? The Temple Mount currently houses the Al Aqsa Mosque and the Dome of the Rock. The Temple Mount is also the holiest place in Judaism, so holy that most Jews believe you cannot walk there for fear of entering the holy of holies, the area where the Second Temple's inner sanctum stood, that holy sanctum being where the High Priest would go only once a year (Yom Kippur), and which houses God's presence. Some small minority however still believe they know where it is: they place it in the Dome of the Rock. The rock is also known as the Foundation Stone, and carries immense significance; it is towards this that Jews have prayed for centuries if not millennia, they face towards it, and that alone should give you an idea of the importance of the land. The Stone is where Isaac is said to have almost been sacrificed by Abraham, among other things crucial to all theology.

• Where is Safed, a city said to have been founded by one of Noah's sons after the flood, with all of its Jewish antiquity sites? In what is today (and would have been in the UN Partition Plan) the northern part of Israel.

• Where is Tiberias, another holy city, founded where one of the earliest biblical villages was, containing the history of Jews in the time of Herod and carries vast importance besides? Also what is now and would have been under the Partition Plan, Northern Israel.

Short of just Christian and Muslim sites, many of them built in areas near or even on top of Jewish holy sites that were destroyed (for example, the Al Aqsa Mosque and Dome of the Rock are alleged to be on top of the Second Temple's remains, which was destroyed by the Romans), and the Second Temple had been built (allegedly) on top of the First Temple, which was destroyed by the Babylonians. Some of the sites hold the same importance for Jews as Muslims and Christians, like the Tomb of the Patriarchs.

It's important to note that Jews don't have to accept Jesus or Muhammad as prophets to have any number of reasons to believe the area contains religious and historical significance for the Jewish people, who prayed in its direction for hundreds of years for that reason, and prayed for a return to it, with the hope of "next year in Jerusalem".