Contest Start represents which day's games will be counting towards your players Fantasy Points.
If the Contest Start says today, that means the games that will be played today will be recorded, as long as the Draft is completed prior to the 'Cutoff Time'. If the Contest Start says Tomorrow then today's games are forfeited and Fantasy Points will begin recording upon tomorrow's game.
In the rare case of the Contest Start having a future dated start, it means that Fantasy Points will begin recording on that specific date.
Contests will typically begin the same day, or the following day of your draft.