The Arc mainly focuses on Monkey D. Luffy, who becomes separated from his crew, and then introduces the Kuja tribe led by Boa Hancock, one of the Seven Warlords of the Sea. Other important characters include Boa Hancock and her sisters, Boa Sandersonia and Boa Marigold, as well as many Kuja warriors like Marguerite, Aphelandra, Sweetpea, Ran, Daisy, and Cosmos.
The main reason this arc occurs is because Luffy gets sent flying to Amazon Lily by Bartholomew Kuma. Amazon Lily is an island with only women, but it is also mentioned to be in the Calm Belt, featuring a rather unique culture and strong warriors.
The Amazon Lily Arc picks up right after Luffy's unexpected landing on the island that no man is allowed to step foot on, where he had caused instant chaos and bewilderment to the Kuja tribe. Now, without his crew and deep in concern for their fates, Luffy has to find a way out of this hostile environment, one that was unseen before, to search for his friends. This Arc will depict a culture secluded in complete fear of men, colliding with Luffy's straight-forward character.
Luffy starts to make them realize that there's actually a person in front of him, be it Marguerite or the other women. However, his presence soon catches the attention of the Island's leader, Boa Hancock. Known as the "Pirate Empress" and one of the Seven Warlords of the Sea, Hancock sees Luffy at first as a threat to her people and her position. So she demands a confrontation in which Luffy faces off against her and her sisters in the arena.
The arc really makes a U-turn when Luffy discovers that he is invulnerable to Hancock's powers because of the Love-Love Fruit. It is this immunity, arising from Luffy's pure heart and lack of lustful thoughts, which arouses Hancock's interest in him and starts changing her impression of Luffy. Also, as the storyline continues, a revelation into Hancock's past is revealed, which explains why she loathes men and is so overly protective of her people. This fact adds dimension to her character and creates a bond between her and Luffy, both having gone through adversity and prejudice.
One of the defining moments in the arc is when Luffy, much to his peril, decides to keep the truth about the Gorgon sisters—Hancock and her siblings—from the rest of the Kuja. This grace and honor action genuinely touches Hancock's heart and consequently changes her disposition towards Luffy. Knowing now what's going on outside the world, more precisely, the execution of Ace, Luffy's brother, the arc focuses on him needing to leave the island and rescue Ace.
Hancock finally agrees to let Luffy leave the island at the very end of the Amazon Lily Arc as a result of falling in love because of his actions and character. This is a great turn of Hancock's character and truly makes her an ally to Luffy. The arc concludes by Luffy's departure from Amazon Lily, now with Hancock on his side, leading into the Impel Down Arc.
The reason this arc is crucial to One Piece is for a few reasons. It gives Luffy some character development because it really shows that even the most hostile people can be won over by his pure nature. The introduction of Boa Hancock and her Kuja tribe serves to expand on the world of One Piece, featuring different cultures and how the complexities work between them and the Seven Warlords system. Finally, the arc acts as a plot bridge between the events of Sabaody Archipelago and succeeding Impel Down and Marineford arcs, setting up the rescue attempt of Ace.