Chilling hours are defined as cool conditions necessary for certain plants to pass through a period of dormancy. Plants requiring chilling hours but are unable to do so (as through abnormal weather phenomena or when planted out of their traditional range) will suffer some degree of distress. Often this means an inability to set or ripen fruit or a failure to grow properly. Blossoms may fail to appear, fruit may be of poor quality and low in yield. Bud break may be tardy and foliage meager or slow to grow.
Temperatures between 32 and 45 degrees F are generally held to be range. Freezing temperatures do not satisfy the period, so when calculating the hours, they hours below 32 degrees F must be discounted. To calculate the hours in your location, the total hours within this period are added. They do not need to be continuous, it is cummulative. There are more complicated models that make distinctions between smaller deviations in temperature and awarding various importance to higher or lower temperatures within the range (ie the Utah model). The term CU or chilling units may be used instead of “chilling hours”, but most (but not all) models treat these terms the same. More importantly is to have a rough idea of the average chilling hours and try not to attempt growing a plant well out of its range.
Now the why. Plants have developed over eons to exist in a particular place and particular climate. Cool season species go through a dormancy which helps it survive a less than satisfactory set of climatic conditions. It goes to sleep to suffer the winter. Preceding this is a hardening off process where the plant slowly engages in physiological changes to get itself ready. It needs that period to be cool without freezing so it can prepare while it is still active. This period varies species to species because they developed in different regions of the planet. Once the requirement is reached, the plant will survive well, and come spring, will emerge in good standing. If the requirements are not, it will suffer.