It is very easy to apply tile grout sealer to your backsplash in an afternoon. The process only requires a minimum of tools, most of which you probably already have or can easily purchase at a hardware store.
First be sure the tile is completely clean so the grout sealer will stick. Clean with any good general purpose cleaner and rinse thoroughly. Allow the grouted areas to dry before starting to seal. Inspect your grouted areas for any loose or crumbling grout and repair with new grout as needed. Be sure to clean the tiles of all excess grout before you start to seal the spaces.
Use your watercolor brush to apply the grout to the grouted areas. Be careful to get the sealant only on the grout. Apply a good thick coat and allow to dry per the instructions on the container. Re inspect your grouted areas to make sure the sealant has completely covered all grout and reseal if necessary.
Wipe away tile sealant spills as you go, do not allow it to dry on your counter tops, or other areas as its difficult to remove once its dry.