This is a common problem when gutters are installed at a later date after the roof and drip edge are finished.
Ideally, the shingles would be extended over to the gutter but that wasn't done here in anticipation of future gutters. One solution is to run some caulking between the top edge of the cutter and the drip edge. Another solution, and a better one which I have done, is to get some flashing from your home store and slide it under your bottom row of shingles and overlap the back edge of the gutter. These are solutions that could have been worked out when the gutters were installed if you knew what to look for or if the installer was looking out for you.