The button called "Edit Layout" at the far left under search space?
I apears you've already edited the layout to make room for the field. And I see that there is a field in Tasks, named Start Date, so it doesn't appear that you need to add a field to the table.
Enter layout mode (such as by clicking Edit Layout) and drag from the field tool down to the top row of this portal. When you release the button, the specify fields dialog appears. Select tasks from the drop down at the top of this dialog. Select the start date field and you have added it to your layout.