Our Gemara on Amud Beis references the principle that only kings from the Davidic lineage were granted the privilege of sitting in the courtyard of the Temple. Most rishonim hold that this is a Torah law (halakha Lemoshe Misinai, see Rashi Sanhedrin 101b), and no other prominent figure is granted this same privilege.
The simple explanation is that a divinely ordained monarchy symbolizes God's glory on Earth and, as such, deserves a similar degree of respect. Much like we saw earlier in Kiddushin 57a, where the commandment to fear God includes a directive to show deference to Torah sages. This seems to be what the Meiri (Sotah 32a) means when he says it is to "show that David's kingdom is the kingdom of heaven.
Rashi (Sotah 40b) says that this is to honor David by acknowledging the completeness of his kingdom. Rashi's choice of words is distinctive, and it appears he was emphasizing a different point. Notes by Rav Hartman on Maharal (Nesiv HaTorah 15:56) explain that sitting implies completeness because one no longer needs to be on the move. Allowing the Davidic king to sit signifies that the pinnacle of human character development has been exemplified and achieved through the kingdom of David. Therefore, the ability to sit in front of God conveys the message that the full aspirations and expressions of mankind's duty toward God have been, or can be realized, through the messianic and Davidic dynasty and mission.
Sefas Emes (Devarim Succos 10) offers a different and powerful explanation for why the monarchs of David were allowed to sit in the temple courtyard, with implications for us regular folk. The Midrash (Bamidbar Rabbah 3) speaks of Aharon initially being chosen by God, then pushed away (due to the sin of the Golden Calf), and finally being brought back and drawn close to God to serve as the high priest and enter the Holy of Holies. Sefas Emes says this is not an accident, but a result of the special status a penitent achieves beyond the righteous, who have never sinned (Berachos 34b). Similarly, David, having sinned with Batsheva and spending much of his life in repentance, benefitted that his descendants were granted a special status, allowing them to sit in the courtyard of the Temple. This implies a certain intimacy granted only to those who have reached such a high level, paradoxically through sin and penitence.
Sefas Emes further states that after the Jews go through the high holy days of Rosh Hashanah and Yom Kippur, they have reached a special status as penitents, granting them even greater intimacy with God. This is symbolized in Succos, akin to dwelling in the Shekhina.
Patterns continue to repeat themselves, spiritually and physically, throughout time and across worlds. Although it's challenging to comprehend, we can intuitively relate to the idea that a relationship that went through considerable difficulty, but ultimately repaired, becomes stronger and more intimate than before. Many couples that I have had the privilege to work with in helping them through painful conflicts, deep anguish, and dysfunction, can testify to this. When they reach that point of deep empathy, care, and repair, all the pain of the past pales in comparison to the special connection and ecstasy achieved through this newfound closeness.
Translations Courtesy of Sefaria, except when, sometimes, I disagree with the translation 
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