The field which will populate the name of the user who is sending messages is {!$} You can get this from the drop-down list while creating templates. Please refer the below screen shots for the same:

Once you select user, you can select any user field from the second drop down which you want to use in SMS Template. After selecting field, click on insert button and automatically that field of the user who is sending message would be rendered when you will use this template while sending out message.