Plant cells are filled with water. This makes the cells firm and causes the plant to be very rigid (plants wilt when they are deprived of water – the cells collapse causing leaves and stems to droop). The pressure of the water inside the plant cells is called turgor pressure.
Some flowers, like Morning Glories, open and close because of changes in the amount of water inside their cells. This water movement through the petals of flowers and the paper flowers you have made is due to capiallary action.
Plants, like paper, have tubes and tube-like structures through which water can move. As the water enters, the turgor pressure produced causes the paper structure to spread open.