The age-up date is the date used to determine the "competition age" of the swimmers for the duration of a swim meet or season.
Season age-up dates
For summer league swim teams, there is generally one date used to determine age for the entire season. This way, a swimmer will always compete in the same age group for the duration of the season. It wouldn't be fair, for example to have a swimmer in the 9-10 age group all season swimming 25s, who turns 11 a day before the championship meet, and then needs to swim a 50 in the 11-12 group for the championship. Using a season-long age-up date avoids this problem.
Season-long age-up dates can be set for an entire Season, found in the Seasons tab under Schedule in the team management console (admin access required).
Meet age-up dates
For multi-day meets, there is the same concern within the meet. You generally don't want a swimmer to swim in a different age group on Saturday than on Sunday because their birthday happens to be on that Sunday. Typically the age-up date would be set to the first day of the meet.
For high school teams, meet events are not generally limited by age, so the age-up date is not important.