Many people when say that their plants are "frozen" seeing the frozen soil they equate this with the plant being frozen but this is not correct. (See: Photo 1)

In fact if a plant freezes it dies. The formation of ice within the cells of a plant is invariably fatal.

Photo 1 (Left):  Escobaria sneedii v leeii
The pot substrate is a solid block of ice, but no damage occurs, in fact this plant is winter hardy up to - 25° C. (Photo by Pál Vajda, Hungary  Nagykőrös)

1. The freezing of air humidity on the plants exposed surfaces (See: hoar-frost)
2. The freezing of the water in the soil.
3. The freezing of "inter"-cellular water in the plant's tissues.
4. The freezing of "intra"-cellular water in the plant's tissues.
    

1) Freezing of the air humidity on exposed surfaces of plants (around 0°C)
The freezing of the water in the air - Hoar frost - occurs at the highest temperature of the four, refers to the ice crystals generally assuming the forms of scales, needles, feathers or fans deposited on plants or exposed objects, that form when the air is moist (saturated) and touches a very cold surface and freezes on it instantly. Hoar frost is often seen on cold, clear autumn nights. Hoar frost can occur at relatively higher temperatures  – usually when the air temperature is around 0° C. However, the ground or exposed surfaces are usually much colder.

2) Freezing of Water in the Soil ( 0 to -5°C)
Water in nature rarely freezes at its physical constant of 0° C as there is always solutes dissolved in the soil water that lowers the freezing point. Frequently a “frozen” ground presents ice crystals only on the soil surface but in this case is not the soil itself that is frozen (it is muddy underneath).
When freeze penetrate deeply in the soil this puts serious and damaging pressure on the roots due to the expansion from the ice. However, plant cells have rigid cell walls and are capable of withstanding several atmospheres of pressure on a regular basis due to their own normal internal turgor pressure. In fact most plants have been shown to survive hydrostatic pressures of over 1000 atmospheres. As the temperature gets lower the plant is in danger of freezing itself , but plants have several mechanisms by which they avoid freezing. One mechanism is the storage of solutes (minerals, sugars, proteins, etc.) within the cell protoplasm. The presence of solutes lowers the freezing point of water and keeps the plant from freezing if the temperature falls below 0° C. In some cases this mechanism provides great protection.

3) Freezing of "inter"-cellular Water ( Approx -5° to -10° C)
Another mechanism that plants use is the expulsion of water from the cell protoplasm. into the intercellular spaces. Changes in cell membrane permeability allow water to leave the cell and enter the spaces between the cells and the water then freezes there instead of within the cell.
In some plant when the temperature falls low enough this phenomenon can be seen, the plant tissues seem like they are "frozen" as they are stiff, icy, almost transparent  and will crack when bent. This is an appearance and the plant itself (living tissue) has not frozen. This mechanism allows the "intra"-cellular  water to avoid being frozen.

4) Freezing of "intra"-cellular Water ( Indicatively -20° to -60° C )
The fourth freezing stage is when the intra-cellular water freezes forming "sharp" ice crystals. This causes the death of the cells that do freeze and is shown through the loss of branches or of the plant itself .
One freezing avoidance mechanism that is not clearly understood is what is called "supercooling". By some means, the cellular "sap" remains liquid at temperatures well below the known freezing point for that "sap". There is some physical relationship to the diameter of the vessels that this "sap" is in, since in larger vessels the "sap" does freeze at higher temperatures. 

Hardiness values
The hardiness value (Frost resistance) for various species only apply to the above ground tissues. The root systems of plants do not undergo the same degree of hardening by any stretch of the imagination. The temperatures at which roots are killed show wide variability from species to species. In nature, plants roots are below ground and are not subjected to anywhere near the widely varying and deeply cold temperatures that the above ground shoots are. When a plant is removed from the ground and placed in a pot its roots can now see temperatures they never saw in nature. This is the very reason why winter protection of container cultivated plant is necessary. Nothing in particular needs to be done to help the above ground parts make it through the winter; it is the below ground parts that need the help. Freezing of the soil in a plant pot is not necessarily a threat to the plant unless it is accompanied by prolonged periods where the shoots are at higher temperatures.

Methods of Protection
Such protection is typically accomplished by keeping the plant in a temperature controlled greenhouse, by keeping the plant in a coldframe, by placing the plant back in the ground, or by mulching around the pot (in order of decreasing effectiveness). All these actions can help prevent the root systems from reaching their "killing temperatures" and prevent root damage or death.

Notes:
Temperate species will enter a period of dormancy in autumn which is triggered by shortening days and/or falling temperatures. Exposure to short days, low temperatures and (very importantly) frost induces the plant to begin "cold hardening" to sustain it from the coming winter temperatures. The above ground parts (shoots, leaves, buds) of many plants from most temperate plants (after "cold hardening") can withstand very low temperatures
The freezing resistance of most plants changes with the season in step with changes in environmental temperature. In summer when temperatures are normally high many plants are more susceptible to a fall in temperature than they are in winter when it is colder. This process,  is known as acclimatization.