On called all-in bets, once all action is complete, all live hands must be tabled. If the all-in action occurs prior to the river, all live hands must be tabled before the remaining board cards are dealt.
In other words, a player can't keep his hand face down, wait for the river, and muck a non-winning hand sight unseen. Essentially, this is the same rule as used in most poker tournaments*, but it is also applied to cash games. I can understand the rationale for the rule in the tournament context—exposing both hands helps prevent collusion and chip dumping (at least blatant versions). But, I cannot fathom a rational reason for enforcing the rule in a cash game setting, where chip dumping is irrelevant, and collusion is rare (at least at low stakes games) and likely not to involve all-ins with any frequency.
This rule has a negative impact on cash game play. Wild bluffs and bad calls are much less common, as players are less willing to push with air, or call with a weak pair or longshot draw, if they face the potential embarrassment of being forced to table their hand. Also, the rule forces players to divulge the range of hands with which they will push all-in or call an all-in in various situations. Knowing that a player will push or call an all-in with top pair, a naked flush draw, etc. is valuable information that players should have to earn, rather than have handed to them by rule.
Now, it's true that the rules in most poker rooms allow players to demand to see any live hand involved in a showdown, though the practice is generally infrequent by custom or house rule. So, the Horseshoe showdown rule merely requires what is already optional. But the Horseshoe version of the rule eliminates an important and established poker custom—the mucked unknown hand—without any compelling rational basis. It is a stupid rule in an otherwise nice poker room, and it deserves to be put out to pasture.
* Tournament Directors Ass'n (TDA) Rule 9:
9. Face Up
All cards will be turned face up once a player is all-in and all betting action is complete.