The first part of the answer is that many did. A huge proportion of the pre-Nazi German-Jewish population emigrated before the war.

I assume you're asking why they didn't all, though. For the most part, they didn't have much of anywhere to go. Palestine had British-imposed immigration quotas (a huge number of German Jews did emigrate to Palestine either within the quotas or illegally), and other countries were not willing to take in large refugee populations.

It's a bit strange to ask why people elsewhere in Europe didn't leave before the Nazis invaded. Sure, the Nazis had clear territorial ambitions for some areas. But even if you do have somewhere to go, leaving everything you know on the chance that the Nazis will overrun where you live would've been a pretty risky choice. Many German Jews left Germany but ended up in other European countries where the Nazis eventually invaded. Anne Frank's family is a fairly well-known example--her family was from Frankfurt, but moved to the Netherlands to find a less hostile country, only to have it be invaded by the Nazis eventually. Most of the Jews killed lived in Eastern Europe, not in Germany or anywhere Germany had obvious territorial designs on. The Nazi threat to them was only clear once the war started, by which time it was too late.

While in hindsight it may seem obvious that the Nazis would conquer huge swaths of Europe, it wouldn't've been so to people in, say, 1935. Emigration can involve tremendous sacrifice, so expecting to have seem mass migration years in advance over the threat of the nearby facists someday successfully invading is unrealistic. Especially when they didn't have anywhere to go.

Believe me, Jews tried. However, there was a meeting of most nations of Europe, North and South America, to figure out how to respond to the problem of Jews seeking to escape the Nazis. It was called the Evian Converence, held in Evian, France, in July 1938. Most nations looked to take their cue from the United States. TLDR: all the countries (except the Dominican Republic) concluded they would accept no Jewish refugees beyond any normal levels of immigrants they already were accepting. The UK accepted 10,000 Jewish children, though. The US didn't even fill the low quota it had. During the war Eleanor Roosevelt begged FDR to admit some distinguished Jewish scientists and artists, which it did. People like Anne Frank's family (which was German) tried desperately to get away. Anne Frank's father's visa applications were rejected everywhere except to the Netherlands, so the family moved there. And then of course were captured and sent to concentration camps a few years later.

A more dramatic example of how no one wanted to admit Jews to their country was seen in the so-called "Voyage of the Damned", when a ship full of Jewish refugees from Germany tried to find a country that would admit them. It did not go well, no one wanted them. And the people on the ship were those who were wealthier, better connected, with better resources. Imagine those who didn't have those advantages?

In addition to the other excellent responses here, it is also important to emphasize that even though a Jew-free Germany was one of the goals of the Third Reich, the German government instituted a number of policies that were counterproductive to this antisemitic goal. One of the fundamental and consistent tenets of National Socialism was that all Jewish wealth was both ill-gotten and deserved to belong to the Aryan Volk. The German government declared that Jewish assets were non-transferable upon leaving Germany and there were a number of "fees" and limits on the amount of money Jews could carry out of Germany. This systematic fleecing of German Jews' wealth limited their options for emigration. Relatively well-off families like the Franks or those with connections to America, which had a number of preexisting charitable networks, had the ability to emigrate. For Jews without these connections or wealth, emigrating out of Germany, let alone Europe, was an incredibly expensive proposition.

Added to this problem of emigration was the context of both the global Depression and an already chilly political attitude towards immigration in a lot of Western countries. The US had already enacted restrictions on immigration in 1924 and the Depression hardened attitudes towards accepting impoverished refugees. The Fascist adventurism in Spain, Ethiopia and Czechoslovakia as well as German rearmament and added a security threat to the economic arguments against immigration.This prompted new security and border controls, which made a valid passport vital for emigration. The Third Reich's position that German Jews were not German nationals and reluctance to issue passports made it much harder for Jews to navigate this system. War jitters could cause countries to renege on preexisting passport regulations, so that some Jewish refugees could fins that their paperwork was no longer in order upon arrival. This was one of the reasons why Shanghai was one of the major destinations for Jewish refugees. The Sino-Japanese War and the collapse of KMT control over the city meant that the Shanghai was one of the areas in the world with little to no passport control.

It is actually quite telling that despite the fiscal and bureaucratic impediments the Third Reich erected to Jewish emigration, many Jews did leave the country.