I just read an article that this is a new feature as of yesterday, January 30, 2018. I would post the link but I am not sure if it is allowed. Currently it is only for Android users (sorry Apple IOS users) and it is only for US Customers at the moment. I am sure they will add other OSes and add International users once they iron out how international and domestic Carriers will benefit from that. Everyone still wants to get paid you know. Anyway according to the article you set it up in your My Profile section in your Alexa App. There is a new toggle button to enable and disable sending an SMS message. You can send a text message to someone in your contacts and someone that is not in your contacts just say the 10 digit phone number then the message.
I have to say I don't see the benefit in opening the Alexa app to dictate a text instead of going straight to my Contacts list on my phone and dictating a text there.
That is because you haven't yet realized you don't have to use the app to do anything on an Echo Device. You are supposed to be using your voice. So there is no need to open any app at all. The same way you say Alexa Call/Drop in on Sabrina and it calls Sabrina for you is the same way you will use this feature. Alexa Send a Text Message to Sabrina. Then you dictate the message all using just your voice. The Alexa App should only be used to add a new Echo Device to your network or household, set up a new feature that has just rolled out. View the new skills that have come out (but you could also just ask Alexa what new Skills are out without using the app too). Using Alexa day to day does not require you opening the app.