If you really don't want to buy it (which I by no means recommend), you could watch the cutscenes or a summary on youtube. HOWEVER, you won't be as immersed as you would be playing the game, and, therefore, not as keen towards the story.
As somewhat of a summary of BBS, Aqua and Terra are tasked with finding Master Xehanort and stopping the dark forces of the unversed. Ventus is warned and prodded by a boy in a mask, Vanitas, to follow Terra and stop him, because Terra won't be the same at the end of it all. Our trio finds themselves all on an adventure, together at heart, but separated physically. Master Xehanort strengthens the darkness within Terra, only fueling the anger Terra holds. Terra meets the masked boy, Vanitas, and learns he is dangerous. At this point, Ven had already met him twice on his journey, and Aqua had met him once. Both Aqua and Terra made it their goal to investigate Master Xehanort, but Terra was mislead, and both Aqua and Terra seek to end this threat of the masked boy. Somewhere along in the story, we are greeted with a flashback of Ven's past that he no longer remembers. Xehanort made Ven his pupil, with the purpose to make the darkness grow within him, but to no avail. Instead, Xehanort separates the darkness of Ven's heart from his light, creating the being of Vanitas. Xehanort, who once craved ever so much to explore the outside "world", lived at Destiny Islands as a young boy, and regarded it as an "empty world". Feeling that Ven is just as hollow inside now, he casts Ven's body to the world, to leave him there. BUT... Ven's heart communicated with a newly born Sora, making an unbreakable connection. This restored Ven back to the light. Xehanort took Ven to be trained under Eraqus, once a fellow pupil, but was stroke by a darkness, a weakness toward the light, resonating in Terra. He would become Master Xehanort's new vessel.
Back to the present. Terra goes back to his home, the Land of Departure. He sees Ven being attacked by his master, Eraqus, and decides to intervene for his best friend, his "brother". He strikes Eraqus down, and fatefully thrusts his homeland into oblivion.
Only heading toward a climatic end, Terra, Aqua, and Ven travel to the place where "keyblade wielders leave their mark on fate", the Keyblade Graveyard. Xehanort's goals were to make Terra's darkness grow and take over Terra's body, and to help Ven grow strong enough (as well as his light) to equal Vanita's darkness and might so that he may create the "X-blade" and summon Kingdom Hearts, which has the power to grant control over all worlds and bring them together as one. The fateful battle begins.
Ven ultimately sacrifices himself to save his friends, destroying Vanitas and the "X-blade". But this only fractures his heart again, and he may never awaken... Terra, on the other hand, failed to foil Xehanort's plans, and is eventfully controlled by Xehanort. Finally, Aqua manages to save Ven's body, and they drift into the Lanes Between (the way of travel to and from worlds). Mickey finds them, and takes them to Master Yen Sid. Yen Sid instructs Aqua to take Ven somewhere safe. Aqua does so, but Ven knows where his heart wants to go. Through the force of his heart, Ven opens a portal to the Land of Departure, though it is in ruins. Seating Ven down, Aqua seals the Land of Departure, transforming it into Castle Oblivion.
Aqua, now with the success of making sure Ven will be safe, seeks out Terra, to free Xehanort's grip on his heart. She finds him at Radiant Garden (Hollow Bastion before it fell to darkness). Beating him, a dark opening begins to swallow Terranort (Terra/Xehanort's combined form), but Aqua acts to save her friend. She instead gets thrust into the darkness, and using her keyblade glider and armor, sends Terranort back up, so that Terra may one day be saved.
The story shifts back to Ven. He again is in his heart. He hears a familiar voice, a warmth that calls to him. It's Sora. Again, he works to connect to Ven, and Ven's heart finds Sora. With Sora's light, Ven's heart may one day be repaired and Ventus may reawaken.
Now the story shifts to Aqua again. However, this is the secret ending, so I will put a spoiler over it if you wish not to see it. [After wandering in the dark realm seemingly forever, Aqua comes to a dark beach. A man sits there. Asking his name, he responds that he does not know. But he does know this is his second time on that beach, and there is a boy fighting against the darkness with his light and his keyblade, saving countless worlds and many others lives. Aqua asks "Is his name possibly Terra or Ven?!" The man says "I am afraid that is not his name." He tells her that he will continue to save and reverse the damage that has occurred, from him and others. He says "His name is..." We then see Namine, Xion, Axel, Roxas, Terra, and Ven, synchronously saying "Sora." The camera shifts back to Aqua, a tear trickles down her cheek, and with a relieved smile, she exclaims "Sora..."]
P.S. The man was Ansem the Wise. Just so you know.(/spoiler)