Aragorn wanted to save the hobbits that he thought were defenseless in the situation (his heroic nature). We also know that he (Aragorn) knew that Minas Tirith was truly his goal -- to claim the kingship. And he seemed drawn in that direction even from the start -- and not drawn toward Mordor, where Frodo had to go. There is no doubt that Sauron was not expecting a lone (or two) hobbits, unaided by an army, to infiltrate Mordor -- and was much caught by surprise when suddenly his end appeared at the Mountain of Doom. Aragorn's presence with Frodo would have put the "radar" up everywhere, and much reduced the chances of success. Aragorn was a lightning-rod in that situation -- his presence would not have allowed the Orcs to write off the capture of Frodo (later on) as merely an unusual spy of no great account -- so Sauron's attention would have been immediately drawn to Frodo if Aragorn had been with him.
It took a great deal of humility and wisdom (great qualities for kings and anyone else) to admit that he could no longer assist the most important aspect of the mission, and to attempt it would only guarantee failure. All his physical prowess and command authority, even as king, would not do the job. All he can do is trust in some unlikely hobbits, while he turned to lesser, but still vital, tasks.
Considering the towering egos (justified or not) of politicians and kings, this was a remarkable action on Aragorn's part.