I don't know why, during Diwali, the Bengalis celebrate Kali Puja? Why don't they worship Goddess Lakshmi and Ganesha?
Not only Bengali Hindus, but Axomiya, Oriya & Maithil Hindus also do the same, following the Tāntrika injunctions.
कार्तिके कृष्णपक्षे तु पञ्चदश्यां महानिशि । आविर्भूता महाकाली योगिनीकोटिभिः सह ।। अतोऽत्र पूजनीया सा तस्मिन्नहनि मानवैः । बलिपूजादिकं सर्वं निशायां क्रियते तु यत् । तत्तदक्षयतां याति कालीविद्या प्रसीदति ।।
Mahākālī manifested on the mahāniśi of the 15th day of the dark fortnight (i.e new-moon day) in the month of Kārtika accompanied by a crore of yoginīs. So She is worshipped on that day by humans. All activities like offering animal sacrifices & conducting worship done in that night yields unperishable merit & satisfaction to Kālī. [Viśvasāra Tantra]
तुलार्के बहुले पक्षे पञ्चदश्यां महेश्वरीं । यथोपचारैः संपूज्य महानिशि नृपो भवेत् ।। शनिभौमदिने चेत्स्यात् ततः शतगुणं फलं । तत्रोभयदिने भूतयुक्तकुह्वां महानिशि । इमां यात्रां कारयित्वा चक्रवर्ती भवेन्नृपः ।।
One becomes a king by worshipping Maheśvarī on the mahāniśi of the 15th day of the dark fortnight when the Sun is in conjunction with Libra (i.e the month of Kārtika) with the pre-required articles. The merit acquired is a hundred times more if the day falls on a Saturday or Tuesday. If this festival is observed on both of those days (of the week) with the mahāniśi occuring at the conjunction of bhūta-caturdaśī with amāvasyā, then the performer becomes an emperor. [Kālīkalpa Tantra]
History Of Kali Puja
The pujas in Hinduism are not based on the Vedas. They are based on individual traditions. Bal Gangadhar Tilak started the Ganapati festival in Maharashtra to foster nationalism and it has been a custom ever since. In a similar way, Raja Krishnachandra who was the king of Navadipa started this Bengali puja of Mahakali in the 18th century. The tradition was carried forward by his descendants and the other wealthy zamindar families in Bengal.
As this puja was basically under the patronage of the rich and famous, it survived on a grand scale. Today, Kali puja is second in grandeur only to Durga puja in Bengal.
The Myth Of Mahakali
There is always mythical story to explain our rituals or pujas. So it is for kali puja. Once upon a time, two asuras, Sumbho and Nishumbho plagued the Earth and the Heaven. Thus the gods prayed to Goddess Durga who is the eternal Shakti of the Universe. To answer their prayers, Kali was born from the forehead of Durga.
Kali attached the errant asuras with a 'khara' which is kind of twisted sword in her hand. Once she started slaying the asuras, her thirst for blood was aroused. She made a garland of the heads of dead asuras and wore around her neck. Then she started slaying anyone who came in her way. To soothe her anger, Shiva (who is her husband) lay in her path. When she stepped on her husband in her fury, she stuck her tongue out in regret. Her fury ended at than moment and that is why she is portrayed with her tongue stuck out and her foot on Shiva.