Why Sarasvati is considered as goddess of knowledge?

Question

In Hinduism, Saraswati Devi is often considered as the goddess of knowledge. Also, it is very common practice to celebrate Vasant Panchami festival (the fifth day of spring) in her honour.

Some Hindus celebrate Vasant Panchami and mark the day, by helping young children learn how to write alphabets on that day.

Source

This clearly shows that Saraswati is considered as the goddess of knowledge as per the faith and common practices.

Below is the how Rigveda mentions about her.

अपो अस्मान मातरः शुन्धयन्तु घर्तेन नो घर्तप्वः पुनन्तु | विश्वं हि रिप्रं परवहन्ति देविरुदिदाभ्यः शुचिरापूत एमि ||

– Rigveda 10.17

Meaning:

May the waters, the mothers, cleanse us, may they who purify with butter, purify us with butter, for these goddesses bear away defilement, I come up out of them pure and cleansed.

Is there any interesting story/event mentioned in Hindu scripture which leads her to be designated as goddess of knowledge?

Answer 1

I'll cite from scriptures to show why Goddess Saraswati is considered as the Goddess of knowledge,wisdom,learning,eloquence,memory , poetic skills etc.

1) The Saraswati Suktam found in Rig Veda 1.3.10 has mentioned(in the last verse of the Suktam) her as the Goddess of learning and eloquence.

2) Next is the Saraswati Kavacham found in the Brahma Vaivarta Purana.Lord Brahma is the Risi(Seer) of this Kavacha.

In this Kavacha Goddess Saraswati is identified with the Goddess of speech(Vagdevi),Goddess of Knowledge(Vidyaadhistatri,Vidya Swarupi),One who resides in prose and poetry(Gadya Padya Nivasini) and one who resides as Vidya in sacred books.

In Phalasruti too, it is stated that recitation of this Kavacha yields memory power,mental sharpness and immense knowledge.

3)The last one is another Stuti of Devi Saraswati done by Rishi Yajnavalkya also to be found in the Brahma Vaivarta Purna.

In this stuti too,Goddess saraswati is depicted as the bestower of Jnana(wisdom),Smriti(memory),Vidya(knowledge) & Kavitwa Shakti(the power of poetry).

She is also depicted as the "Adhistatri of all Vidyas" and as "Akshara Swarupa"(the Personification of alphabets).

She is depicted as the embodiment of Smriti Shakti,Jnana Shakti,Buddhi Shakti and Kalpana Shakti and also as the ruler of tongue,mind,thought,speech and anything to do with brains.

Conclusion-Goddess Saraswati is considered as the Goddess of Knowledge due to facts that are firmly rooted in our scriptures and not because of beliefs.

Edit-

That Goddess Saraswati is to be worshipped on Sukla Panchami Tithi of the Magha month is mentioned in scriptures.

Mentioned in other Puranas as well but here i'm quoting from the Devi Bhagavatam as found on this page:

Sage Narayana described the methodology of worship of Devi Sarasvati, the powerful manifestation of Mula Prakriti, the Goddess of Knowledge, Mental Faculties, Memory, ‘Mantras’, Literature and Fine Arts. Her symbol is ‘Veena’ (Lute) and Books; Her ‘Vahana’(Vehicle) is ‘Hamsa’ (Swan) and Her male counterpart is Para Brahma. On every Sukla Panchami of Magha month and on the day of commencing education, the worship of Sarsvati is to be performed as per Kanva Sakha Yajur Veda hymns, if possible.

Answer 2

She was indeed born like that from left half portion of Brahma. Many Puranas such as Shiva Purana, Matysa Purana describes this birth story of Sarasvati. But Brahmavaivartara Purana describes slightly a different version of her birth. It says Sarasvati was born from left part of Dharma who was born from chest of Sri Krishna. Brahmavaivartara Purana is Krishna- centric text and it says Krishna is Supreme Brahman.

सौतिरुवाच
आाविर्बभूव कन्यैका धर्मस्य वामपाश्र्वतः। मूर्तिमूर्तिमती साक्षाद् द्वितीया कमलालया॥ ५३॥
आविर्बभूव तत्पश्चान्मुखत: परमात्मनः। एका देवी शुक्लवर्णा वीणापुस्तकधारिणी॥५४॥
कोटिपूर्णेदुशोभाढया शरत्पङ्कजलोचना। वह्निशुद्धांशुकाधाना रत्नभूषणभूषिता॥ ५५ ॥

Then Sauti said : “At that point of time a beautiful girl emerged out of the left part of Dharma who resembled completely with Laksmi. She was known by the name of Mūrti. Thereafter, a female with white complexion emerged out of the mount of the lord, holding a book and a lute. She had the lustre of crores of suns. Her eyes resembled the lotus flowers of the winter season. She had bright costumes and was adorned with the bright gold ornaments.

सस्मिता सुदती श्यामा सुन्दरीणां च सुन्दरी। श्रेष्ठा श्रुतीनां शास्त्राणां विदुषां जननी परा।। ५६ ।।
वागाधिष्ठातृदेवी सा कवीनामिष्टदेवता। शुद्धसत्त्वस्वरूपा च शान्तरूपा सरस्वती॥ ५७॥
गोविन्दपुरतः स्थित्वा जगौ प्रथमतः सुखम्। तन्नामगुणकीर्तिं च वीणया सा ननर्त च।५८।।
कृतानि यानि कर्माणि कल्ये कल्ये युगे युगे। तानि सर्वाणि हरिणा तुष्टाव च पुटाञ्जलिः।। ५९ ।।

She wore a serene smile on her face and she had beautiful teeth. She was the damsel of sixteen year’s age. She was the best of all the beauties in the universe and all the Srutis, Šāstras and the intelligence emerged out of him. She controlled the speech, the goddess, of all the poets, an incarnation of purity and was known as Sarasvati with a peaceful look. Standing before Krsna, she played on Wind and then danced. She sang in praise of the lord, highlighting all the chief events of his earlier incarnations.

So, Saraswati manifested as Goddess of Knowledge, music, and speech. There's no particular story of acquiring these qualities or skills.