Yes, the phone has to be on, and it must be connected to a cellular network and have a working data connection for MightyText to work properly.
Your phone has to be on because MightyText routes SMS messages VIA your phone.
This is how MightyText can send messages that show to your recipient as coming from your phone number. Your phone number is assigned to you by your carrier/operator and is linked to your phone itself.
If you are looking for a solution that will work when your phone is off, we would recommend trying a product like Google Voice - however keep in mind that SMS will not integrate with your Android phone, and it will be a different phone # from your Android phone.