I'm answering these questions in a slightly different order, as this is what flows best for the story of Hebrew, in my mind.

So, how did Hebrew go extinct?

Long story short, Aramaic beat Hebrew. This wasn't something Hebrew specific, Aramaic started to become the common language of the Near East from 900 BC, becoming the diplomatic and administrative language of the Assyrians, the Achmaenids and the Seleucids, its status not being reduced by the use of Greek, Akkadian etc. The Aramaeans (who spoke Aramaic) migrated across the Near East, spreading their language as they went, many Aramaic speakers being conquered by the Assyrians, becoming a part of good old Mesopotamia. The Arameans- like the Jews- did not flee the conflicts following the collapse in the Bronze Age as one, instead they spread out, radiated even, in small groups and communities. This ensured the Aramaeans' survival and spread their influence- along with Aramaic.

So by now, the Assyrians had adopted Aramaic as their administrative language, simply because so many Aramaeans had been absorbed, and Aramaic became the vernacular language. Aramaic was soon being adopted all over the Near East, spreading around Judaea. The Babylonian captivity was when Aramaic became especially important, around 6th century BCE. From then on, intellectual and educated Jews were usually multilingual, speaking Greek, Aramaic and Hebrew, with Hebrew being used as a vernacular language until the end of the 2nd century.

And like how French was historically seen as prestigious due to its international and diplomatic use, Aramaic became the language, the language of status. It was enough of a status symbol, and was used enough in administration too, for locals to start using it natively.

Why did Hebrew develop?

The revival of Hebrew in the late 19th century coincided with the rise of Zionism. Eliezer Ben-Yehuda was the founder of Modern Hebrew, although he certainly came from a background where Jewish identity as something high, as something not just meaning the shtetl you were raised in, but literature, the arts, etc, was becoming more recognised. Jewish writings in Yiddish were becoming more popular and more famous, and Hebrew was also seen as important to this high Jewish identity, with a few authors writing in (biblical) Hebrew.

So why did Hebrew develop then? Zionism and movement to Israel really triggered it.

1. It was a language which crossed ethnic and geographical boundaries within the Jewish communities. Jews have historically spoken many, many languages, from Judeo-Tajik to Yiddish to Ladino to Judeo-Arabic to Krimchak. Their Jewish identity and Judaism linked them together, and Hebrew was an integral part of practicing Judaism (and still is). Although some people thought using Hebrew in every day speech might be sacrilegious - using a holy language, purely associated with religion, to ask to go to the toilet in school, for example- it was a language that, when Israel was founded, served to connect the Jewish diaspora together, and provided a 'base' language for which to communicate in. Someone who is Polish might be friends with someone who is Turkish, but if they live in England, they'll probably communicate in English, as their base language, their shared one. Israel did not have such a language, for the Jewish immigrants at least, so the recently revived Hebrew was used as one.

2. Hebrew was also used because diasphoric languages were looked down upon. To use Yiddish is to reject the uniting language of Israel, a reminder of the fact that a Jewish diaspora existed in the first place! There was a lot of backlash against it, with Yiddish speakers being intimidated in the streets, and Yiddish language entertainment being censored or not getting any audience due to the dislike of Yiddish as a symbol of the diaspora's existence. This was not limited to Yiddish at all, although Yiddish was the most prominent and had the most active cinema, literature, theatre etc scene compared to Ladino, Judeo-Arabic and so on. Yiddish would also have been most prominent in many of the founders' minds, being primarily Ashkenazim, so coming from a Yiddish background. For example, Ben-Gurion, who's native language was Yiddish, disliked the use of Hebrew. Hebrew was truly revived by Eliezer Ben-Yehuda, who was a Yiddish speaker and Ashkenazi. His son, Itmar, born 1882, could be described as the first native speaker of Modern Hebrew.

3. Hebrew as a liturgical language has been used for three thousand years, straight. Religious leaders had to knew Hebrew inside out and many lay people also could read Hebrew, for their religious and rabbinical studies. It survived like Latin has survived in the Vatican. It had not died out and was a constant across Jewish cultures, and was closely connected with religion, so it won favour in certain religious groups (although its close connection also caused a lot of objections, using the Holy Tongue for daily speech).

The rise of Zionism was not simply wanting to return to Israel. Cultural Zionism, as Asher Grinsberg called it, was also an integral part of the movement, with Jewish cultures becoming more important, not something to be ignored or looked down on. Jews were not wanting to assimilate, they were wanting to revive Jewish cultural practices. Israel was going to be a vibrant nation, not just one inhabited by Jews, but a living embodiment of Jewish culture. Israel needed to be Jewish, with the culture that was a part of Jewish identity, not merely a country holding a significant number of Jews.

One way of reviving Jewish cultural identity was through Hebrew. Asher Grinsberg wrote in Hebrew, talking about his aim of reviving and revitalising Jewish culture. Of course, there was a problem, in that many Zionists, including those of similar beliefs to Grinsberg (eg, not Herzl!) were Ashkenazi, and Yemenite culture, Indian Jewish culture and so on was often overlooked,

So it was from this background- a time where people were placing increasing importance on the idea of Jewish identity and culture, a time of Zionism and a kind of nostalgia for the Israel of the past- that Eliezer Ben-Yehuda decided to revive Hebrew. He did this in a number of ways, and he also faced opposition. He set up language classes, raised his children to speak Hebrew as their native language (and only home language), set up a Hebrew newspaper, to promote the revival of Hebrew.

From the 1900s onwards, Hebrew gained a steady foothold on the Zionist movement. Yiddish- the language for a significant percentage of Zionists- was diasphoric, and it's absorption of various Germanic and Slavic elements symbolised the exile. It symbolised everything Zionism wasn't considered to be about. It held a story of Jewish lives far away from Israel, mixing and integrating and taking on practices from Gentiles. Some Zionists rejected Yiddish and other diasphoric languages as a result of this symbolism.

So why was Hebrew revived? 1) Hebrew being a base language for a heterogenous nation 2) Hebrew being ideologically in line with Zionist ideals and 3) Hebrew was an expression of the growing high Jewish identity.