Start by cleaning the interior and exterior of your entry door. Use only a mild soap and water mixture. In some cases, warm (not hot) water will even do the trick without any soap. Do not scrub or use harsh abrasive cleaning chemicals or materials. Don't forget to clean inside the door frame as well. Dirt loves to hide in small spaces so you might find a soft toothbrush handy as well.
Once your door is clean, you can then take a look at the weatherstrip all the way around the door. If you notice any damage or excessive wear, you should replace the parts of the weatherstrip or all weatherstripping if needed.