The answer lies within the two brothers who started the lines in the first place. When Elendil came across from the sinking Numenor, he brought with him his sons. The eldest of which was Isildur, and the younger was Anarion. Together they created the two kingdoms of Gondor in the south, that was entrusted to the rule of Anarion, and Arnor in the north that was given to Isildur to preside over.
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Elendil maintained supervision over both kingdoms until he died in the Battle of the Last Alliance, in which a band of elves and men defeated Sauron and prevented him from rising to power. Without the protection of their father, the original founder of the kingdoms, Arnor quickly fell apart and fractured into 3 kingdoms known as Arthedain, Cardolan and Rhudaur. All 3 of these quickly met terrible fates, as Cardolan was overtaken by war and famine, Rhudaur fell to the evil command of the Witch King out of Angmar.
Meanwhile, Gondor was being passed from generation to generation in relative peace, until it came into the hands of Ondoher, who is the king that the beheaded statue with the crown of flowers is thought to be based on. Ondoher died valiantly in battle, but without having produced any male heirs, which left the throne of Gondor up for grabs. The current king of Arthedain (Arvedui) quickly married Ondoher’s daughter, and claimed rightful kingship over both kingdoms.
However, disputes against his claims arose, and Gondor decided instead to elect a recent war hero by the name of Ernil, who was a distant cousin of Ondoher, because they believed that his might and valor (plus his small link to the bloodline) would make for a much better king. Unfortunately, Angmar’s reach spread, and Arvedui (who is still a descendant of Isildur’s direct line) was killed, thus ending his rule in Arthedain.
Arvedui’s son survived and went into hiding, and it is his line that continues directly all the way down to Aragorn, making him Isildur’s heir. Earnur, Ernil’s son was the last king of Gondor, and after the valiant reign of him and his father, the throne went unclaimed for several years. Various people tried to re-take the throne across the ages, but all were denied, for it was felt that if even the heroic and legendary kingship of Ernil failed, surely there could be no other that would do a better job, and therefore the kings were doomed to fail and throw the country back into turmoil, exposing the people to uprisings, invaders and usurpers. It was thought safer to hand the throne into the protection of the stewards, who would guard it without blood and ancient lineage getting in the way. This is how it comes to Denethor, and then to his far more noble son Faramir.
Many Lord of the Rings fans believe that Aragorn’s father Arathorn may have been able to re-claim the throne, had he lived long enough. He was a brave and skilled warrior, as well as a humble and caring man, and may have shaped up to be the kind of king that the people of Gondor would accept and love. But alas, he died when Aragorn was 2, at the hands of a poisoned orc arrow, leaving the small child and his mother to flee to Rivendell, where Aragorn was taken into the protection of Elrond.
When his mother died, Elrond raised Aragorn like a son, and encouraged him to pursue a path that would make his worthy of sitting on the throne of Gondor, with the love and respect of his people, rather than trying to claim some ancient birth-right or rule by force. This is how Aragorn became involved in the Dunedain Rangers and met the Grey Company, on his quest to improve his skills and knowledge, travel the lands, and become the savior that people needed.
Thus, after the war of the ring when Aragorn does eventually reclaim his rule of the White City, he has earned the ability to do so through his valor in battle, and his healing of the people with an ancient art that only the kings of old can recreate. Minas Tirith is rebuilt under his loving guidance, and his kingship becomes legendary among men for the peace and thriving that it brings throughout all of Middle Earth in the Fourth Age.
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