These are following Walkthrough of mission Cabmaggedon in Grand Theft Auto:Vice City

First enter the marked taxi in Kaufman Cabs to trigger the cutscene and start the mission. Delores dispatches you to pick up Mercedes Cortez in Viceport.

Head to Viceport in the cab. Baby your wheels to keep them intact, or drive it like you stole it - there's a Pay n Spray near Mercedes's icon. Either way, you'll want a pristine cab to pick up Mercedes.

Drive up to the icon (in the middle of the lot, near 8-Ball's garage) and you'll be prompted to honk the horn for Mercedes. Do so, and trigger a cut-scene: six Vice City Cabs stream in through the gate, and the gate closes behind them. You've been set up, and these guys are here to kill you.

Passing this first part of the mission requires you to survive the onslaught of the six cabs until the timer runs down. Note that your car doesn't have to survive: only you do. You have a few options for doing this:

1. Evade the six cabs. This can be done, but it's not easy. Try to get the cabs grouped together and then avoid the group, otherwise you'll have to keep track of six cabs in six locations.
2. Destroy the cabs. Drive-Bys are the most obvious way to do this, but they're not the only way. If you've got some heavy weapons, like the Minigun or Grenades, that's another possibility.
3. Drive into 8-Ball's Bomb Shop (one of the last blue garages near the water) and the cabs will follow you in. After the door closes and opens again, your cab will be rigged to explode. Exit the car and jump over the swarming cabs, then exit 8-Ball's. Detonate the cab-bomb and all the rival taxis will go up with it. (Courtesy of the BradyGames Official Strategy Guide)
4. Leave the area and let the timer run down, the cabbies kill themselves, or both. A low wall separates the lot from the water. Hop onto this (be careful not to fall into the water) and its height will provide you some protection from the VC Cabs cars. With very good luck, you'll be able to destroy the cars and kill off any surviving drivers. More likely, though, one of the cabs will manage to launch itself into you, knocking you into the water and failing the mission, not to mention killing you.
Better is to head along the wall away from 8-Ball's, toward the shipping containers. From here, you can either climb up onto your car, if you parked it strategically, or the VC Cabs, if you're lucky, and climb up onto the the shipping containers. The cabs can't reach you here, and again, you can pick them off at your leisure. Alternately, you can let them destroy their cars and explode.
Or, you can walk along the low wall past the shipping containers, into the rest of the City of Vice City. Grab a health pickup and some armor, and head back to the lot. The timer runs out and/or the cabbies self-destruct, and you're through this first part of the mission.

Once you're through, another cut-scene triggers: "Time for Kaufman Cabs' guardian angel to eat some fender." A tiger-striped cab enters the lot through the gate, which once again closes. This guy has to be killed off - there's no reconciliation possible. If you're feeling patient, you can wait him out. Eventually he'll wreck his car and it will explode. Sometimes he's smart enough to get out of the car before it goes, though, so you may still need to finish him off. Doing so will give you his melee weapon: a katana.

If you're not feeling patient but are feeling lucky, a sniper round to the head will kill him in the driver's seat, leaving you his car. Otherwise, machine guns or heavy weapons are more sure. 8-Ball's bomb shop is also here, and you may be able to take him out with your car. However you choose, kill him to pass the mission, opening the gate, completing Kaufmann Cabs' missions, and causing the tiger cab to spawn in Kaufman Cabs.

Note: If you exit the lot as in option 3, it is possible to steal a heavy vehicle while the six original cabbies are still running around trying to get to you. Driving this truck around to the gate will place you on top of where the tiger cab spawns, causing the cab to spawn under you. This will either destroy the cab as it spawns (killing the driver and passing the mission), destroy the truck and possibly kill you (failing the mission), or flip the truck.

Also Note: if you leave the lot, or even are hanging out by the shipping containers when the tiger cab spawns, the cab may have difficulty finding you, causing you to have to seek it out, likely on foot. Be armed and be armored.