If you haven't seen this deck before, it will look pretty strange. It doesn't play lands and spells like a normal Magic deck—instead, it chooses to draw first, immediately discards an eighth card to the cleanup step, and proceeds to move cards from the library to the graveyard as quickly as possible using the dredge mechanic to replace its draws.
Once you've generated some free creatures courtesy of Narcomoeba, Ichorid, Nether Shadow, and Bridge From Below, you flashback Cabal Therapy to strip the opponent's hand of answers and then sacrifice three creatures to Dread Return a Balustrade Spy. You target yourself with the ability, milling your entire library (since there are no lands in it), and proceed to Dread Return a Flayer of the Hatebound and then a Golgari Grave-Troll to shoot the opponent in the face for somewhere around 40 damage.
The phases of the turn sort of serve as your spells, and free cantrips from Street Wraith and Gitaxian Probe generate more dredging and speed up the whole process. Playing this deck is unlike playing any other deck in Magic; you're playing a fundamentally different game than the opponent, so they usually just sit there doing random irrelevant stuff while you grind them down with free abilities and inexorable Zombies. You're not so much piloting a deck as herding a horde of mad monsters toward your opponent—just let your cards loose and they'll do the work for you. All you have to do is hit your triggers and figure out what to name with Cabal Therapy.
The most important and often overlooked card in the deck is Phantasmagorian. This unplayable creature from Planar Chaos is the best possible discard on turn 1, allowing you to dump your whole hand to its last ability and kick start the dredging process. You can discard either three or six cards to it (in the opponent's end step before you dredge for the first time on turn 2) by activating its last ability twice; you discard three, hold priority, and discard anther three with it still in the graveyard.
This is a common line of play that leaves you with only Phantasmagorian and one other card in hand, ideally a Gitaxian Probe to cast and try to win the game on the second turn. Discarding Phantasmagorian rather than a card with actual dredge also allows you to play around Deathrite Shaman sniping your single creature out of the graveyard on turn 2 (as can cycling Street Wraith in response to Deathrite's ability if you don't have a Phantasmagorian handy). Later in the game, you can often get two copies Phantasmagorian going to ensure you always have a Golgari Grave-Troll in the bin for maximum dredging.
You can do a fancier trick with Phantasmagorian too if you have a Street Wraith and another cantrip you want to keep in hand. Say you have Phantasmagorian in your bin, Gitaxian Probe, Street Wraith, Golgari Grave-Troll, and four other cards in your hand on the opponent's end step before your second turn. The play here is to discard Troll and two other cards, hold priority with Phantasmagorian in the grave, cycle Street Wraith dredging Troll back to your hand, and then discard the Troll and two other cards to activate Phantasmagorian again.
This leaves you with Phantasmagorian and Probe in hand and Troll in the bin ready to be dredged in the draw step! The sheer amount of cards you can dredge with this opening makes for a very powerful start, and every time this play came up during the Open, I won on the next turn.
One general tip to keep in mind when dredging is to put Nether Shadow as low as possible in your graveyard. Although you cannot change the order of cards already in your graveyard in Legacy (thanks to the existence of cards like Nether Shadow), whenever you dredge a group of cards, you can choose the order they get put into the bin. This means whenever you're dredging or using Balustrade Spy, make sure to sort the Nethers to the bottom of however many new cards you're putting into the grave to maximize the chances of having three creatures above them.
Figuring out whether to use Ichorid and which creatures to exile is one of the few times playing this deck is more art than science (the other main one being Cabal Therapy). In general, I err on the side of always reanimating Ichorid since if you dredge into a Bridge from Below during the draw step, the upside of getting a Zombie token when Ichorid dies is very high. Street Wraith is the primary food for Ichorid since it doesn't do anything once it's in the bin. Any copies of Balustrade Spy after the first one are fairly useless too, as is Golgari Thug once you have a few better dredgers in the grave, so those are the next tasty morsels you should go after with Ichorid.
Get hasty 3/1 beats in wherever you can (and remember Nether Shadow has haste too), but usually try not to attack into any board where the opponent can trade or chump with a creature and exile your Bridges. Also remember that Bridge exiles off of any creature dying on the opposing side even though it only makes Zombies from your nontoken creatures dying. Bridge From Below has my vote for most confusing card played in Legacy, so read it, learn its secrets, and know it inside and out before taking Dredge to a tournament.
When casting Cabal Therapy, remember that all you have to do upon casting it is sacrifice a creature and declare a target player; you don't name a card until it resolves. Also remember to trigger Bridge Zombies before your Therapy or Dread Return resolves since otherwise you can be considered to have missed them. I never Therapyed myself during the tournament, but that play sometimes comes up if you need to get a dredger or Balustrade Spy into the yard, so remember that is an available play as well.
Mulliganing
Don't do it. I've used this deck for 3 years and I've never mulliganed, which I doubt any other (non-Dredge) player can say. Of course things get dicey if you don't have a dredger in your opener, but when you only need one instead of the usual 2-3 lands, it's easier to find . Even if you're missing dredgers, I think it's usually correct to keep, knock the deck, and trust in your first draw step plus Gitaxian Probe and Street Wraith to miracle the missing piece. Every mulligan literally sets you back a turn, so once you pick up this deck you lock yourself in to keeping basically any seven-card hand. We're not here to play Magic with our friends—we're here to keep seven, move cards from pile A to pile B, and unleash Zombies on fools.
Sideboarding
The pool of cards that function in a deck without lands is fairly tiny, so trying to scrape together fifteen more after you've already built a 60-card deck is pretty rough. I built a sideboard that is basically just answer cards for other fast combo decks, a package for Show and Tell, and wild card Sickening Shoal that I bring in against anything with creatures. It's not pretty, but with a maindeck this perfect, who needs to sideboard?
Ashen Rider is fairly self-explanatory; it's just the best answer to Show and Tell that's also black and a sick reanimation target on its own. You can go fairly deep Dread Returning and Cabal Therapying the Rider between the graveyard and battlefield to bust up lots of permanents. Blightsteel Colossus looks a bit weirder since you obviously can't reanimate it, but it is another piece of the Show and Tell package. Having it in your deck allows you to Show and Tell a Balustrade Spy on their turn; mill yourself down to just Avatar; untap; and win if they pass the turn with their Emrakul, the Aeons Torn or Griselbrand or whatever.
Sickening Shoal is the card I boarded in the most, usually just as a straight swap for Mishra's Bauble or Shambling Shell against all the non-combo decks like Jund and Delver. Again, it only functions in the opening hand, so you don't want too many of them, but it can be clutch in the right situation as an answer to Deathrite Shaman or a creature equipped by Umezawa's Jitte (both of which came up during the Open). It also gives you a desperation out if you're locked under an Elesh Norn, Grand Cenobite and forced to actually draw cards looking for an answer—luckily Phantasmagorian has a CMC of seven, perfect for Shoaling Elesh Norn or Griselbrand. It's possible a second one could find its way in the deck.
Mindbreak Trap and Surgical Extraction are the most powerful sideboard cards, giving you interaction in the early game against things like Storm, Tin Fins, and Belcher. Noxious Revival usually acts as a fifth Surgical versus Reanimator type decks, but you can get fancy by recycling Narcomoeba back on your deck or fizzling opposing Surgicals.
Remember folks, sideboarding is all about who you usually play against. As the format changes, so will both of your boards. I always look for new ways to keep this deck updated and will continue to do so.