Stop rain water from dripping between the gutter and fascia board.
Drip edge flashing is used at the eave or edge of the roof to protect the fascia board from water damage and rot.
Unfortunately it is often installed wrong.
Your first indication of drip edging installed wrong is the rain water that is dripping from under the rain gutter, and the dirty streaks and stains across the gutter board. If the gutter itself is loose or buckled from improper installation, there may also be debris, ( leaves twigs and such ) sticking out from under the gutter.
Rain water splashing up from the gutter will seep in between the drip edge flashing and the rear flange of the gutter and drain down the gutter board.
Besides the ugly stains on the gutter board, water that gets behind the rain gutter can lead to peeling paint, rusted nails and fasteners and rotting wood.
You can resolve this problem in two ways. One, have your roofer remove and re install the drip edge over the gutter. Or, have a gutter specialist remove and re install the rain gutter behind the drip edge flashing.