If I understand correctly, your table is working correctly by storing a checkbox value of checked vs. unchecked, but your form doesn't have the checkbox, but rather a text box showing True/False values. Is that an accurate statement?
If so, you have the wrong control type. In the design view of your form, you'll have to use the design tab and select the checkbox control from the controls menu on the ribbon. Insert the checkbox on your form and then use the Properties menu to link it to your Field 1 on your table. This shows the results in a checkbox format rather than a True/False text box.