Why Goddess Durga was worshiped on dusshera?

Question

Dusshera is the day the Ravana was killed by the Rama, but Hindus worship Goddess Durga instead of Rama on dusshera ? Why so?

Answer 1

In West Bengal, Assam, Odisha, Durga Puja is also called Akalbodhan.

Both the words Akaal and Bodhan are Sanskrit words, which are also included in many other Indian languages, such as like Bengali and Hindi. The word Akaal means untimed (kaal=time and a=not) and the word Bodhan means worship or invocation. Thus, Akaal Bodhan means worship or invocation of Durga in an uncustomary time. It is given this name since the period of this worship differs from the conventional period, which is during the spring (Basant).

Akaal Bodhan

In the Ramayana, Rama goes to Lanka to rescue his abducted wife, Sita, from the grip of Ravana, the king of the Demons in Lanka. Before starting for his battle with Ravana, Rama wanted the blessings of Devi Durga. He came to know that the Goddess would be pleased only if she is worshipped with 108 Neel Kamal (blue lotuses). After travelling the whole world, Rama could gather only one hundred and seven of them. He finally decided to offer one of his eyes, which resembled blue lotuses. Durga, being pleased with the devotion of Rama, appeared before him and blessed him. The battle started on the Saptami and Ravana was finally killed on the Sandhikshan (i.e., the crossover period between Ashtami [the next day] and Navami [the day after]). Ravana was killed on the tenth day, Dashami. Since the period of this worship was different from the conventional period, this puja is also known as Akal-Bodhan; a worship (Bodhan) in an unconventional time (A-Kaal).

More details here

Answer 2

The anecdote of how shardiya Navratri started, is mentioned in Kalika Puran. The purpose of akalbodan or untimely invocation of devi was to ensure the destruction of Ravana and victory of Shri Ram Chandra and this is closely related to Uttarayan summer solistics.

The verses from Kalika Puran,Chapter called, Mahishasura upakhyan states as follows,

In the olden times in order to kill Ravana and bless Sri Rama, Brahma Dev did the invocation of Devi at night. After that she gave up on her resting phase and enter into the Nanda tithi of Shukla paksha of Ashwin Maas. Then from where she was been worshipped by Rām, she went to Lanka. Soon after going there, she allowed the Ram and Ravana to start the duel. The battle between Ram and the Army of Ravana lasted till the Saptami Tithi. The devi was thus worshipped for the 7 days by devta to ensure complete victory of Ram. And finally Ram killed Ravana on the Navami Tithi. Thus, she was again worshiped by all deities on Navami. The lord of all worlds, Brahma himself did this special Puja of Durga Devi. Thus with Sarvarotsav, the Devi was bid farewell on the tenth day, that is on Dashmi.

Shri panchanan Vedant Shastri, explains that,

""Since Bhadrakali was in the favour of Ravana Sri Ramachandra wasn't able to slay him. It wasn't good time period for Shri Ram. At that unfavourable time "Akal", Shriram invoked devi at her own instructions."

3rd Skandha, Chapter 30 - On the narration of the Navarātra ceremony by Nārada and the performance of that by Rāma Chandra

41-42. Vyāsa said :-- Then the powerful Bhagavān Hari, hearing all from the Muni, believed them to be true; and, on the approach of the month of Āśvin, prepared the altar on the top of a hill and placed the Auspicious Goddess, the World Mother and, observing all the rules, performed the vow and worshipped the Goddess.
43. Fasting for nine days, Rāma celebrated the vow and duly offered sacrifices, performed the worship and Homa ceremonies.

On explaining the importance of Ashwin Navratri various shlokas can be found.

Durga Saptashati, 12th Chapter-

शरत्काले महापूजा क्रियते या च वार्षिकी।
तस्यां ममैतन्माहात्म्यं श्रुत्वा भक्तिसमन्वितः॥12॥
सर्वाबाधा*विनिर्मुक्तो धनधान्यसुतान्वितः।
मनुष्यो मत्प्रसादेन भविष्यति न संशयः॥13॥
During autumnal seasson(Ashwin/Shardiya), when the great annual worship is performed, the man hearing this glorification of mine with devotion shall certainly through my grace, be delivered without doubt from all troubles and be blessed with riches, grains and children.

Devi Bhagwatam 5th Skandha, Chapter 34 - On the methods of the worship of the Devī.

In the bright fortnight of the month of Āsvin or Caitra, is to be observed the fasting of the Navarātra by those who desire for their own welfare.
Whoever celebrates the holy Navarātra ceremony in the bright fortnight of the month of Āśvin with his heart full of the thought of the Devī, gets all his desired fruits. 

Thus amongst all 4 Navratris Shardiya Navratri holds the highest place.

कालिकार्पणमास्तु ।

Answer 3

In the great war between Rāma and Rāvaṇa, the goddess was also involved. Here are the verses from the Kālikā Purāṇa.

In olden days the great goddess was waked up by Brahmā when it was still night in order to favour Rāma and to get Rāvaṇa killed. Then having given up her sleep she on the first day of the bright half of the moon in the month of Āśvina went to the city of Laṅkā, where Rāma stayed in the past. The great goddess Ambikā having gone there caused Rāma and Rāvaṇa engaged in a battle, but she herself remained hidden and kept eating flesh and drinking blood of the demons and the monkeys. - Kālikā Purāṇa 60.25-28

She got Rāma and Rāvaṇa engaged in fighting for seven days, and when the seventh night passed away, the goddess Mahāmāyā, the embodiment of the world caused Rāvaṇa to be killed by Rāma on the ninth day. During the period of seven days when the goddess herself observed the game of the battle by the two, she was worshipped during all those seven days by the gods. On the ninth day when Rāvaṇa the hero, was killed, Brahmā along with all the gods performed a special worship of Durgā. There after the goddess was dismissed on the tenth day with the celebration of Śāvorotsava. - Kālikā Purāṇa 60.29-32.

The same incident is also narrated in detail in Mahabhagavata Purana.

Reference-

Kalika Purana English translation by B. N. Sastri