Felt is a membrane that is fixed between your roof timbers and the tiling material. Overtime, the felt will wear. Fitted correctly, the felt will be dressed over the fascia into the gutter. Problems occur when the felt dressing into the gutter deteriorates. This can lead to water dripping down the fascia and wall causing damp problems.
There are a couple of tests you can do yourself to check the state of your felt. If you have access to your loft, climb inside safely and turn the light off and close the loft hatch. (Be very careful if you are doing this). Look towards the eaves of the roof, if you can see light entering in anywhere you may have an issue. The other test is simply to stand beneath your guttering next to the house wall and look up at the guttering. If you can see the bottom of a tile then there is a problem with the felt.