Instagram Partner Messages is an important setting that has to be set up in order to start building automations. Usually, this setting can be accessed in your Instagram app but it might so happen that this setting section is not available to you because of Instagram UI changes. But no worries: it's an easy fix.
Alternatively, you can easily access this setting within Facebook Business Page Settings.
Go to your Facebook account and choose a Business Page that is connected to the Instagram account you are trying to connect.
Then go to the Business Page Settings --> Instagram.
In this section, you will have to disconnect your Instagram account and then reconnect it again.
After you click 'Connect Account', you will be asked to Allow Access to Messages (this is the Partner Messages function that needs to be toggled on).
Click 'Continue' and log in to your Instagram account to complete the step. You are good to go.
Finally, go back to ManyChat and try to connect your Instagram account once again.