If the transactions come in automatically, Tiller puts the dates in automatically. If you bring in and/or make transactions manually (which is what I assume you mean by import), then you need to enter the dates yourself.
The date in the Month column is always the first day of the month. So, if the transaction date is 05-01-19 through 05-31-19, then the Month column date is 05-01-19.
The date in the Week column is the day of the week beginning on Sunday. So, if the transaction date is 09-18-19, then the Week column date is 09-15-19. So, if your transaction date is 09-15-19 through 09-21-19, then your Week column date is 09-15-19.