Is the door black on the outside and white on the inside?
trickle some water from a kettle or teapot onto the sill and into the trough. It should drain away from the doorway, away from the house, towards the outside, and thence towards the road, without leaving puddles or running towards the door. Does it?
there should be drainage holes in the trough, possibly at the sides. use a hoover or small brush to remove any dirt from the trouch. You may need stiff wire to unblock the drainage holes.
If it is draining correctly, you can add a weatherbar to throw any water than runs down the surface of the door outwards, to handle rare occurrences of driving rain onto the door in strong winds, but my first thought is that the drainage and slope are not correct. This sometimes happen if the slabs or tiles outside the front door are too high or sloping the wrong way, for example when they are added after the door has been fitted, or laid on top of a previous paving.
I can see some kind of bracket on the external wall. is there a small roof? It should direct water away from the door but may be running off and splashing up. Stand back and take a wide shot.