It seems that you True / False values are stored as text and not as booleans. But only boolean cells can be exported to boolean database column. You can check actual type of a value by opening right click menu of a cell and choosing “Metadata” menu item.
You can convert them to actual booleans with the “Modify column” action and one of the following expressions.
If you sure that all the values in the column contain either ‘True’ or ‘False’:
[Bool] = 'True'
If you want to check data quality:
if([Bool] = 'True' or [Bool] = 'False', [Bool] = 'True', error("Value '" & [Bool] & "' can't be converted to boolean"))