Take it that the start of the week is on Monday. And, today is Thursday. So, last week can be anytime between last Monday and last Sunday.

I've been here since last week. = I arrived here sometime last week (could be anytime between last Monday and last Sunday) = X days plus number of days up to today

I've been here for the last week. = I arrived and stayed (been here for) a whole week (seven days) up to now. (7 days plus number of days up to today)

Do you follow that?