Before you do any appliqué project, you should be sure to have a pair of appliqué scissors on hand
Appliqué scissors are medium-size scissors with a weird duckbill on one blade.
They are specifically designed to trim appliqué fabric after you have stitched it down, and before sewing over the raw edges of the appliqué fabric with a final satin stitch.
The unique design of the these scissors allows you to trim very close to the stitching line without cutting a hole in your base fabric.

The proper way to use appliqué scissors is to insert the duckbill side of the blade between the fabric you want to cut and the fabric underneath that will remain uncut. This enables you to cut the appliqué fabric close to the placement stitching without inadvertently cutting the fabric underneath.  The duckbill will be underneath the pointed blade with the duckbill between the appliqué fabric and the background fabric and the pointed blade on top of the appliqué fabric.