Our amidah/shmoneh esrei follows a precise structure: it opens with praise of Hashem, in the middle are requests and supplications, and it closes with thanksgiving. We normally do not voice requests in the opening brachos -- one must first acknowledge Hashem's dominion and majesty; only then is one in a position to ask Hashem for personal needs. Yet, the tefilos between Rosh haShana and Yom Kippur prove an exception to the rule. The Rishonim are already troubled by the fact that zachreinu, mi kamocha, kasveinu l'chaim, and b'sefer chaim, the special insertions made in tefilah to ask Hashem to inscribe us for a good year in the book of life, all appear in the first and last brachos of tefilah, not in the middle where personal petitions normally are made. Why do we depart from the normal pattern?
R' Avraham Bloch (Shiurei Da'as, "Ori v"Yishi") explains that the special insertions made this time of year are not like other personal requests. The recognition of Hashem as king of the universe, as master of all that occurs in the universe, ideally also entails the recognition that we exist solely to help proclaim those truths. We ask zachreinu l'chaim not because we selfishly want a few more years in the world, but rather because in this holy time of year we hopefully are contributing or hope to contribute to Hashem's presence being felt in the world. We say to Hashem that not only is his presence felt in the world because he is rofeh cholim, he is matir asurim, he is someich noflim, but also because he will be zochreinu l'chaim and through our existence will his presence be felt to a greater degree in this world.