When I run the print material requirements, there is a field called beginning MRP balance. Sometimes this number has 3 asterisks after it. I can't seem to understand where this number comes from, because the ones with the asterisks make no sense to me.

The *** is an alert that the OH quantity has changed since the MR was generated.  So you don't get caught making decisions on an out-dated MR run.  It's saved me a few times.

That is one possibility.  There are other things that can cause it.  
 
The MRP Report "On Hand" is the total on hand quantity for the item for all Locations at the time the report is run.  
 
The MRP Beginning Balance is the calculated on hand from the MRP records for the beginning of the report.  This could be limited to only certain Warehouse Locations if the MR-F generation was run for only selected Locations.  It could be as of some other date if the report is run with a filter on Start Date.  Or it could be as Dick mentioned, there has been transaction activity and the current UOH is different from what it originally was when MR-F was run.