How to Sew Up "Cut Outs", (Shapes Cut Into Fabrics)?
Here are some additional details on the sewing and gluing methods mentioned in my first answer:

Sewing a Satin Stitch
either by hand or by machine: a satin stitch is lots of horizontal stitches (vs. vertical) all very close together.



Here is a YouTube video showing both methods of satin stitching, by hand and by machine: https://www.youtube.com/watch?v=_eTurIxJlFg  

Note:  the video shows how to sew (fasten) cut out letters onto another piece of fabric; you are cutting out your letters from the piece of fabric.

Satin stitch along the outline of your letters before you cut the letters out.



Using Fabric Glue (one brand is Fray Check): 
use the fabric glue bottle like a pen, squeeze a thin line along the outline of your letters.

Let the glue dry for about 15 minutes, and then cut your letters out.  

Note:  I’m not sure what the iron is for (the advice you got at the store).  You don’t need to use an iron with a fabric glue.

Hope this helps.  Good luck with your project!