There's lot's going on here. Remember that when you are using a tarp, a groundsheet will prevent any water in the ground (dew fall, dampness, water table etc) from evaporating and condensing on the inside of your tarp.
Now, since you pitched late in the day when there was very little evaporation going on, the dampness this morning, on the inside could have been water evaporating this morning and hitting the cold tarp sheet, and the dampness on top is just the dew from last night. Under the tarp should be slightly warmer than the surrounding air and this morning would have heated up quicker in the rising sun and any moisture hitting the cold tarp condenses and is wet.
However, there's lots of other factors, like is the tarp 100% waterproof, was it frosty etc
It would have been a lot worse if you were under it unless there was a very draughty through flow of air. That's partly the reason why inner tents (tarps!) were created, to keep the condensation from you.