Red shirting is actually easy to explain. College athletes have 5 years to complete 4 years of eligibility. When an athlete redshirts, he chooses not to compete for his school for a season, saving a year of eligibility for later. The term "redshirt" comes from the old days of football, when members of the team who were sitting out the season would sit in the stands wearing red shirts.
Athletes can redshirt one season at a time. For example, they could compete during cross country, then redshirt indoors, and compete again outdoors. Generally, competing once in uniform during a season means the athlete waives the right to redshirt. If the athlete becomes sick or injured early in the season, he may apply for a medical redshirt to maintain the year of eligibility. This is rare but does happen.
Athletes redshirt for various reasons. Many freshman athletes in sports that require physical maturity (track and football are the main ones) redshirt to allow them to catch up with older, more developed athletes before using their eligibility. In theory, they will be much better athletes in their 5th year of college than their first year.
Sometimes, coaches choose to redshirt athletes for strategic purposes. If a coach knows his team will have a bad season, he may redshirt his top athletes to set up a much better year in the future. Athletes have also redshirted in Olympic years so they could focus on training without the need to compete for their schools.
Redshirted athletes are still allowed to race, but only in non-scored meets and not in their team's uniform. They are also allowed to practice with their teams and may also receive scholarships.