Usual Suspect Inscribed thus:

Maytag LSE7800.

The H & C supply lines go to the mixing valve and a single hose goes
from there to the tub. But before it connects to the tub there
inserted in the hose a plastic cube with some kind of restrictor. The
cute is open at the top. Open?

Yes, it's a 5-sided cube with no top. The water enters one side and,
apparently, flows out to the tub.

What's the name of this part?

It is not called out in these drawings:

http://snipurl.com/maytag.lse7800.diagrams

What's its purpose? It seems to be the cause of some local flooding in
the laundry room. What I thought was a cracked hose seems to be,
instead, this cube overflowing.

How does this part function and how do I determine if it needs
replacing?

Thanks,

If its not part of the soap dispenser, then its an "air break" ! Its
designed to stop contaminated water getting back into the water supply.

If its flooding from the air break then either the incoming water supply
is too high or the is a partial blockage at the tub end. Normally it
would never need replacement. However I have seen them get cracked and
leak.

HTH

--
Best Regards:
Baron.