The humidity in your tent is rising at light off because the temperature has dropped, the amount of water it takes to saturate it with moisture becomes less.
So for the exact same moisture content it'll read 50% at 25c but 65% at 20c and 80% at 15c (the figures are only an example you'd need to Google the exact % moisture to temperature)
Relative humidity is a measure of the moisture saturation for a given temperature, frosty cold air has very little ability to hold moisture so a tiny amount of moisture will make it read 100% saturated, hot air can hold lots so it takes many times more moisture to saturate it.
The dew appearing on the grass is due to heat disappearing and the air no longer having the ability to hold moisture so it dumps it, clouds do the very same thing.