Cut flowers can stay alive for a short period of time because they are still drawing up water. Water normally enters a plant through the roots. There is a one-cell layer thick "skin" on roots, called the epidermis, that soaks up water present in the soil. The roots also contain tiny hairs that absorb minerals from the soil. The water and minerals combine in the center of the root, then flow up into the stem. This process of upward flow is called "transpiration pull" and is possible due to the process of evaporation and the strong attraction between water molecules. Flowers that are cut are abruptly severed from this support system, and it sends them into a shocked state.
Flowers respond to the shock of being cut by sealing off the tissue in their stems to protect the water that remains in the stem. An air bubble may also form in the bottom of the stem. This is why it is important to re-cut a flower stem just before you place it in water. Otherwise, it will not "drink" new water due to the sealed tissue or air bubble. Placing a cut flower in water will allow the flower to stay alive for a while longer because flower is still getting water. Instead of water being absorbed by the roots, however, the water is moving directly into the stem. The flower is not getting any minerals, however, unless plant food is added to the water.
Over time, cut flowers gradually stop drinking water and die. This is due partially to bacteria attacking the cut stem and partially due to the fact that the flower is no longer getting nutrients and minerals from the soil. Even if plant food is added to the water, it is not enough for the cut flower to survive. As water stops flowing to parts of the flower, those parts will wilt. Eventually, the stem will also wilt as it ceases to draw up water.