And how exactly should XL determine that the date was overdue?

Take a look at Conditional Formatting in XL Help.

If, for instance, the due date is in A1 and the date is overdue if it's
in the past, select A1, choose Format/Conditional Formatting... and use
something like:

CF1:        =A1<TODAY()
Format1:    <patterns>/<color>

Look at conditional formula with a condition of Formula Is and a formula of
=A2<TODAY() and set the colour.

Use format>conditional formatting, then formula is and

=A2<=DAY()

click format button and select red pattern or fonts and click OK twice

where A2 is the cell with the date you want to check if it is overdue
if it is 30 days from the date in A2 you can use

=A2+30<=TODAY()