Rosh HaShanah is the day the world was created and Hashem became King. Every year on this day we coronate the King anew. Why is Rosh HaShanah a day of judgement? The pasuk in Mishlei says, "Melech BaMishpat Yamid Eretz", the main function of a King is to maintain order in the land. Therefore the first order of business is to see where each citizen stands and what will be his fate.
It is brought down in the Zohar and other places that one should not make personal requests on Rosh HaShanah, for it is a disgrace to the King. Our entire focus should be on the glory of the King and His coronation. We should concern ourselves with what we can do for Hashem, and not what we'd like Hashem to do for us.
Yet Rav Yisroel Salanter and most gedolim today say that we may make personal requests, for in our generation most of us are not on the level that we can truly say that we spend the day in awe of Hashem's glory. At least by asking for personal requests we show that we acknowledge that Hashem is the King, and to Him we turn for help.
Rav Leib Chasman explains with a Mashal. On the day of the King's coronation a royal banquet was held. All the ministers were present, and the magnificently decorated hall inspired awe, glory, and fear. During the ceremony a pauper dressed in rags, who somehow managed to sneak by security, runs up to the king and starts screaming that he has no food to eat and he is desperate for some potatoes for his family. The guards pounced on him at once and removed him. Yet even as the ceremony resumed the mood and atmosphere had soured and the prior glory was not restored.
In a similar but very different version of the story the pauper throws himself at the King's feet and says my master my king, my family has been impoverished and suffering for years. We have tried every avenue to extract ourselves from this situation to no avail. Your are the all powerful, and we know you to be kind and merciful. We are all your loyal subjects, and you and only you can help us. Our lives are in your hands alone, and we give our entire being to your service. Please have mercy!
Beyond all the glamour of the event, this moment is the true coronating moment of the king!