There are several factors that come into play when you are talking frost and how it forms, of course temperatures, dew points, wind speed, humidity are all important but the most important one is the Wet Bulb Temperature.
The Wet Bulb temperature is the temperature air cools to when water is added in the equation. We are all familiar with this phenomenon. You feel colder when you are 40 F and wet, than when you are at 40F and dry. This is similar to how your skin cools when you sweat and you step out into the wind. The temperature on your skin is cooler than the ambient temperature around you.
Wet bulb temperature is almost always lower than air temperature. In many cases it is considerably lower than air temperature. When water is applied to a plant, the plant surface temperature initially drops to the wet bulb temperature, even when air temperatures are well above freezing.
Here is a chart showing the difference between the two. Notice the number of hours below freezing where frost can occur.
Basically this chart shows that the air temperature can be above freezing and at the plant surface you can have freezing temperatures and that is how frost develops.
Walking or driving on the turf when frost is present can severely bruise the turf. At this time of year we will suspend any play on the golf course until the frost has melted or has burned off.