Rosh Hashanah is known as the “head” or the “king” of the new Jewish year. It is therefore appropriate that on this day we declare Hamelech! – that Hashem is our King. Yet there is no command in the Torah that requires us to be oleh regal, (to ascend to Yerushalayim) for the holiday of Rosh Hashanah as we do on the other three Torah-ordained holidays. Wouldn’t it be appropriate for every person to come in the presence of Hashem, the King of all kings, on this solemn and tremendous Yom Tov?
I would like to share with you some answers that I saw in the Torah Tavlin. The Sfas Emes gives an explanation based on the Gemara Rosh Hashanah (26a): There are two things that are not permitted in the Beis Hamikdash due to the principle of אֵין קַטֵּיגוֹר נַעֲשֶׂה סַנֵּגוֹר (a prosecuting attorney cannot become a defense attorney) and they are: wearing the golden vestments when the Kohen Gadol performs the Yom Kippur avodah(service) and blowing the horn of a cow as a shofar on Rosh Hashanah.
Although the Kohen Gadol enters inside the Holy of Holies for the avodah, while the shofar is blown outside, the Gemara still equates the two. This is because the sound of the shofar is so effectual that Hashem “remembers” Klal Yisroel because of it, and bestows mercy upon us; thus, it is considered as if it was performed “inside,” as well. For this reason, we do not actually need to go up to the Beis Hamikdash on Rosh Hashanah; by properly performing the mitzvah of tekiyas shofar (blowing of the shofar), it is deemed as if we are already בְּפָנִים, as we are already inside the Beis Hamikdash.
There’s an answer from Rav Moshe Wolfson shlita, who explains that regarding the Aseres Yemei Teshuvah (the ten days of repentance), as it says: Search out Hashem where He can be found; call Him when He is near (Yehoshua 55:6). It is a time when Hashem is easily accessible, more so than at any other time of the year. Thus, there is no need to make the pilgrimage to Yerushalayim for Rosh Hashanah; on Rosh Hashanah, Hashem comes to us!
This is why on Rosh Hashanah the chazzan shouts “Hamelech” from his place in the shul and only then makes his way to the amud (the pulpit). This custom symbolizes that on this day, Rosh Hashanah, we do not need to travel far to be close to our Creator – wherever we are, we call out to Him – and He comes to us! The Mezritch Maggid would say: “Even a peasant who has a message for the king from a long-lost son, is promptly ushered into the throne room. All the more so, the prayer of a person who does teshuvah is precisely such a message and is warmly and promptly welcomed by Hashem.