The story is described in the Srimad Bhagavatam:
Hearing of the beauty, prowess, transcendental character and opulence of [Krishna] from visitors to the palace who sang His praises, Rukmiṇī decided that He would be the perfect husband for her. Lord Kṛṣṇa knew that Rukmiṇī possessed intelligence, auspicious bodily markings, magnanimity, beauty, proper behavior and all other good qualities. Concluding that she would be an ideal wife for Him, He made up His mind to marry her.
But Rukmini's brother Rukmi hates Krishna, so he stops his family from giving Rukmini to Krishna, and makes arrangements for marrying his sister to Krishna's enemy Shishupala. So Rukmini sends a Brahmana to Dwaraka to deliver the following letter to Krishna:
O beauty of the worlds, having heard of Your qualities, which enter the ears of those who hear and remove their bodily distress, and having also heard of Your beauty, which fulfills all the visual desires of those who see, I have fixed my shameless mind upon You, O Kṛṣṇa. O Mukunda, You are equal only to Yourself in lineage, character, beauty, knowledge, youthfulness, wealth and influence. O lion among men, You delight the minds of all mankind. What aristocratic, sober-minded and marriageable girl of a good family would not choose You as her husband when the proper time has come? Therefore, my dear Lord, I have chosen You as my husband, and I surrender myself to You. Please come swiftly, O almighty one, and make me Your wife. My dear lotus-eyed Lord, let Śiśupāla never touch the hero’s portion like a jackal stealing the property of a lion.... O unconquerable one, tomorrow when my marriage ceremony is about to begin, You should arrive unseen in Vidarbha and surround Yourself with the leaders of Your army. Then crush the forces of Caidya and Magadhendra and marry me in the Rākṣasa style, winning me with Your valor. Since I will be staying within the inner chambers of the palace, You may wonder, “How can I carry you away without killing some of your relatives?” But I shall tell You a way.... If I cannot obtain Your mercy, I shall simply give up my vital force
So to sum up, this was a rescue, not a kidnapping. Rukmini calls this a Rakshasa marriage, which is where the bride is willing to get married but the bride's family isn't, so the groom fights off the bride's family and whisks the bride away. This is to be distinguished from a Paishacha marriage, where the bride doesn't want to get married either; the Paishacha marriage is the one strictly forbidden in the Manu Smriti. The Rakshasa marriage is allowed for Kshatriyas, but arguably this isn't even a Rakshasa marriage, because Rukmini's parents were also consenting to the marriage; it was just Rukmini's brother who was objecting and making the army of Vidarbha oppose Krishna.