How did the Jewish people leave their country?

Answer 1

It depends on the country in question, but there are two ways that Jews commonly left the countries they were living in: forced exile, or persecution fierce enough to compel flight. Jews changing countries of residence and citizenship by choice is relatively recent phenomenon.

Answer 2

The Diaspora (scattering of the Jewish people) began because we were unable to live in the Holy Land. The prophets had constantly warned the Israelites that ignoring the Torah would result in exile.
1) Around 2600 years ago, the Assyrians forcibly exiled the Ten Israelite tribes to points unknown. A small percentage of each of these tribes is still among us, but most of them were exiled and didn't return.

• Link: Where are the Ten Tribes
  

2) Around 2500 years ago, the Babylonians destroyed the First Temple and forcibly exiled the remainder of Israel's population to Babylonia.

• Link: The Destruction
  

While the Jews were permitted to return to Israel (Judea) seventy years later, and tens of thousands did so (and rebuilt the Temple), most of them remained in Babylonia, while others began to settle in North Africa, southern Europe, the Crimea, throughout the Near East and elsewhere.

3) In 68 CE, the Romans destroyed the Second Temple. The Romans did not force the Jews out of Judea in a single expulsion. Rather, the Romans expelled them from Jerusalem only; and the rest of Judea lost its Jews slowly, over a period of centuries, as living there became too harsh. Even then, we have records of Jewish communities who lived in Judea (Palestine) during the entire period of the last two millenia.

• Link: History of the Jews in Israel
  

Those Jews who left Judea went to southern Europe, North Africa, Arabia, the Near East, and (slowly) further afield (especially throughout Europe).

• Link: Jewish history timeline
  

Related topic:

The diaspora (scattering) was difficult for us:

1) Because certain mitzvot (such as those related to agriculture in the Holy Land) now became suspended for the duration of the exile.

2) Because of the impediment to communication between the various communities. This could lead to the development of rivaling groups, each claiming to be authentic.

3) Because the Jews were now a minority among other nations, who were sometimes quite hostile.

4) Along with the Torah, the Land of Israel (Judea) and the Holy Temple had been central to Judaism. It now was to be seen whether Torah-observance could be maintained on the high level that those ideal factors had been conducive to.

Related topic:
How did the diaspora affect the Jews?
To a certain extent, the diaspora caused the various Jewish communities to take on minor aspects of their host countries.
The religion itself was affected little, if at all, since all Jews share the same Torah and Talmud, and differences within halakhah (Jewish laws) are relatively small. In addition, the Jews were dissuaded from assimilating because the Romans were the polar opposite of Judaism; plus the fact that some of their greatest historians spewed diatribes of hate against the Jews.

• Link: Contrasting Romans and Jews

However, in secondary matters of culture such as pronunciation, mannerisms, and introduction of new vocabulary, every community is influenced somewhat by the others among whom they dwell; so some traits of our regions of residence have rubbed off on us. Examples are the differences in pronunciation and accent (for the Hebrew alphabet), differences in dress, and minor differences in customs.

See also the other Related Links.

Link: Maintaining Judaism during the exile
Link: Where do Jews live today?