Not really a history question.

But here's a free translation: "To neither fall victim to the past nor the future. It is all about being in the present."

There are a few grammatical constructs in there that don't translate well into English. But the meaning is relatively simple.

source: I speak German

To add a bit of context to the preface, Jaspers was one of Arendt's mentors in her education and the two maintained a correspondence throughout their lives. But Arendt's choice of a Jaspers quote was not due to their professional and intellectual relationship. Jaspers was one of the more prominent German public intellectuals in the postwar period. The Nazification of the German university system he lost his position at Heidelberg for his liberal politics and marriage to a Jew, but he was otherwise not directly molested by the regime. This put Jaspers in a prime position in the postwar period to speak about German culpability for the crimes of Hitler. The American military government assigned him as one of the intellectuals to rebuild intellectual life after Nazism. Jaspers's 1946 book Die Schuldfrage dealt with the issue of German guilt for those like him: people that had not directly participated in criminal acts, but did not resist the regime. Jaspers castigated trends of applying collective guilt to a select few groups like the SS or Party members as lazy and unreflective. For Jaspers, guilt was both a collective issue and a personal one; Germans needed to both interrogate both their personal (in)actions and what the Third Reich's actions meant for German national identity.

Jaspers's books and writings created a good deal of controversy in postwar Germany, especially since the dominant trends in German memory politics favored either castigating a select clique of Nazi offenders or some loose sense of collective guilt that would be handled by a successor German government and functionally meant nothing to daily life. Jasper's approach the question of guilt though was in resonance with Arendt's approach to the modern dictatorship. She argued that people needed to examine how they served as cogs within a modern system which had the latent ability to turn into murderous tyrannies. Within this context, the Jaspers quote makes more sense in that he contended the best responses for Germans who lived through National Socialism was to try and make the experience relevant for the construction of a new, better Germany.

Also, it should be anheimfallen, not aheimfallen.