Several existing answers have already answered why people from the 24th Century would have learned hand writing.

However, in reference to the episode that brought you to this thought, there is a reason why Chakotay specifically would be familiar with hand writing.

Chakotay is of Native American descent, and although reluctant to learn at times, was still taught the Native American ways growing up. From the Memory Alpha Wiki on Chakotay:

Being of Native American descent, Chakotay's tribe – mainly because of the intrusion of more technological societies – left Earth to find their own home on another planet near the Cardassian border. From an early age, his father tried to impart his values on Chakotay in many ways, such as taking him on hikes to nearby forested worlds of their ancestors.

And more information about Chakotay's tribe and the planet he was raised on can be found on the Memory Alpha Wiki page on Dorvan V:

In 2350, a group of Native Americans from Earth settled on Dorvan V, having searched for a new home since 2170. The settlers were worried about losing their traditional culture on an increasingly culturally-homogeneous Earth, and this was the main reason for their quest. As a result, in the 24th century, the population still lived much like their ancestors did many centuries before.

No doubt Chakotay would have been exposed to and been taught handwriting in his "traditional" upbringing, more so than other humans would have.

Several existing answers have already answered why people from the 24th Century would have learned hand writing.

However, in reference to the episode that brought you to this thought, there is a reason why Chakotay specifically would be familiar with hand writing.

Chakotay is of Native American descent, and although reluctant to learn at times, was still taught the Native American ways growing up. From the Memory Alpha Wiki on Chakotay:

Being of Native American descent, Chakotay's tribe – mainly because of the intrusion of more technological societies – left Earth to find their own home on another planet near the Cardassian border. From an early age, his father tried to impart his values on Chakotay in many ways, such as taking him on hikes to nearby forested worlds of their ancestors.

And more information about Chakotay's tribe and the planet he was raised on can be found on the Memory Alpha Wiki page on Dorvan V:

In 2350, a group of Native Americans from Earth settled on Dorvan V, having searched for a new home since 2170. The settlers were worried about losing their traditional culture on an increasingly culturally-homogeneous Earth, and this was the main reason for their quest. As a result, in the 24th century, the population still lived much like their ancestors did many centuries before.

No doubt Chakotay would have been exposed to and been taught handwriting in his "traditional" upbringing, more so than other humans would have.