How do players quickly get to Mephisto?

Legitimately fast Mephisto runs are the result of a few things:

1. The Teleport skill. This is either a skill that a Sorceress will level up in her skill tree, or a skill gained by an item such as Enigma runeword armor.
2. Faster Cast Rate. A player with multiple items giving Faster Cast Rate can cast Teleport very quickly, and as a result, cover more ground quicker while searching for the next level. Items like Wizardspike give this benefit. Here is one of many charts which you can find online to show you where you gain the most benefits from Faster Cast Rate, with an explanation here.
3. Searching clockwise. Depending on who you ask, you either search the map clockwise or counterclockwise. These maps are generated randomly, so learning a few map layouts will not help you. I loaded up the game and did one quick search on Hell difficulty, traveling clockwise, hugging the wall(s):
4. A bit of an exploit. Characters like Sorceresses will often use this trick to kill Mephisto easily and quickly. It involves luring him to one of the moats of blood just South of him, the player getting to the other side of the moat, and killing Mephisto from a location which he cannot get to or attack.
   A player such as a paladin with Enigma runeword armor who teleported to Mephisto can easily stand beside him and kill him, and will never have to use this exploit.

And that is about it for how to find Mephisto as quickly as possible using legitimate means... You just have to manually search each map.

Now... The more likely scenario for people completing Mephisto runs quickly is that they are running a hack (cheat) program that will help them or do it for them. There are two ways for this to be implemented by perhaps unsavory players:

NOTE THAT THESE METHODS ARE AGAINST THE TERMS OF AGREEMENT AND CAN RESULT IN YOUR CD KEY BEING PERMANENTLY BANNED FROM DIABLO II'S ONLINE SERVICES IF DETECTED

1. There are maphacks which reveal the entire map, meaning that a player will know exactly where to go the moment they enter the level. These are still in heavy use as of 2012.
2. There are bots which players can run. Bots can automatically control a player's character, navigating them to a boss/ monster and killing it for them. These are usually set up to even create new games and continually repeat this process. These are also still in heavy use as of 2012.

It's all about patterns. First of all, teleport is mandatory to get through the Durance of Hate in any reasonable amount of time. Especially on Hell difficulty, when the Durance is so big, teleport is absolutely required.

The main memorable thing about the pattern is that the stairs down to the next level of the Durance will always be at the edge of the map. This doesn't mean exactly what you would expect*, but in 95% or more of cases, traveling the perimeter of the map using teleport is enough to find the stairs. With faster cast speed and chugging mana potions if necessary, it is a fairly quick search.

Obviously, once you get to Durance of Hate level 3, the layout is always the same and teleporting to Mephisto is essentially a straight line.

*The caveat to this is when the map wraps around itself strangely, the "edge" of the map could technically appear relatively in the middle of the whole minimap. This is a rare layout, but something to keep in mind. If you search the whole circumference of the Durance and still haven't found the stairs, it's worth looking at the map in more detail quickly so that you can find where that "inner edge" might be and direct your search.

For Mephisto, the rule for quick, legitimate runs is to use the clockwise rule.

That doesn't mean blindly traveling clockwise around the perimeter looking for the stairs down.

The Clockwise Rule When Running Mephisto

What the clockwise rule means is that if you look at the room that the waypoint is in, and where it joins to the rest of the level, then rotate that room clockwise in your mind, you'll know where the exit room attaches to the map.

Teleport to the edge that you'd expect the room to attach to as quickly as possible and search for the down stairs. If you don't find it on that edge of the map, look for other places that a room could attach at that angle.

90% of the time, you'll also be travelling clockwise to cover the most likely edge, but occasionally the starting room will attach at an unusual location, and you'll be more likely to find the exit fast by teleporting straight across the map to the opposite edge, or travelling counter-clockwise, or even (albeit very rarely) exploring the same edge that you started on.

Because the ending room can also end up in unusual locations, occasionally you'll end up exploring most of the level before you find it.

The General Clockwise Rule

The clockwise rule applies in almost every underground level in Diablo, though there are a few places where you might consider traveling counter-clockwise instead, if you want to skip a waypoint or side-dungeon, and a very small number of exceptions, such as the Worldstone Keep level 2.

In general, the following elements are arranged clockwise: entrance, way-point, side dungeon/gold chest, exit. Each of these will be rotated clockwise from the previous.

If any is missing, the remaining ones will (usually) be arranged in clockwise order as listed. Most levels don't have all of these, and the exit is a clockwise rotation from the entrance.

In a few cases, notably the Catacombs and the Worldstone Keep, you can't tell the rotation of the level from the entrance...you can tell from the waypoint in Catacombs level 2 what the rotation of the exit to that level will be (clockwise), but once you reach Catacombs level 3, you have to hunt for the exit to level four and Andariel with no help from the entrance.

Similarly, in the Worldstone Keep, you can tell the rotation of the room with the down stairs from the rotation of the room with the waypoint on level 2 (as mentioned, it's an exception, the exit is counter-clockwise), but you dont get any clue as to the rotation of level 3 from the entrance, and have to search for the exit.

Aside from Mephisto, the clockwise rule is most helpful when running the Countess, since the entrances there do tell you the rotation of the exit.