Had these gutters installed last year. During a rain storm, water runs behind the gutter and down the fascia. What good is my drip edge doing, if it is BEHIND the gutter? 
Were these gutters installed correctly
gutters
Related Solutions
Sounds like either the gutters aren't secured tight enough, or you don't have a drip-edge flashing between the roof and gutter. The drip edge deflects water out and away from the house trim and into the gutter. You install it under the shingles, pulled out slightly from the trim, and secured with roofing cement.
Hardness is a good way to check wood for damage without pulling it apart. Take a nail and press it into the wood where you know the siding is in good shape. Then, do the same where you suspect damage. If the wood is softer, then you can assume it has some amount of water damage. If you want a more direct, verifiable method your best bet is to yank a shingle or two off and check behind them. (From the pictures it looks like you have shake but its hard to tell.)
Also, as Steven commented...remove those "gutters" as soon as is humanly possible and replace them with actual gutters. Each year you wait things will get exponentially worse. If there is not currently water damage (which I doubt, sorry) there will be very soon.
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Best Answer
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.