The wind is caused by the difference in air pressure or atmospheric pressure. The bigger the difference the faster and the stronger the winds. Air moves from high pressure to low pressure.
The passing of a front will cause the winds to pick up. The air-mass behind a cold front is cold and the air-mass in front of the cold front is warm. The colder the air mass the lower the pressure is. The warmer the air-mass the higher the pressure is. The larger the temperature and pressure gradient before and after the cold front the stronger the winds will be. Here are a few examples below.
If the temperatures ahead of a cold front are in the 80s and behind the front, they are in the 40s the winds will be quite strong. The reason why is because there is a sharp temperature/pressure gradient before and after the cold front.
If the temperatures ahead of a cold front are in the 60s and behind the front, they are in the 40s the winds will not be as strong. The reason why is because the temperature/pressure gradient before and after the cold front is not as strong.
You may be thinking what does the temperature have to do with air pressure? Well, the warmer the air the higher the pressure, and the colder the air the lower the pressure. The bigger the temperature gradient the bigger the pressure difference. Remember, the greater the pressure difference the strong the winds.