I listed champions that can use this build, it doesn't mean other champions can not use it.
This guide is just about why people (also in high ELO) get these items and how it works. It looks stupid because you would think that items like Ravenous Hydra, Bloodthirster, or Blade of the Ruined King are core items on certain champions and there is no way why you would leave them (in tournament you never see this build on Zed who is the only champion of the listed champions who is played in tournament recently). This build is very situational and can go very wrong.

It's all about early-mid game damage, it's about killing people very fast which is not possible as fast with other items.

Do these items even stack?

Yes. UNIQUE means that you can not stack two items with a UNIQUE passive of the same kind, 2 Brutalizers, 2 Ghostblades or 2 Black Cleavers don't stack with each other. As long as the passive is unnamed and just called "UNIQUE Passive" you can stack it with every other passive as long as it's on a different item. UNIQUE passives with the same name like "Butcher" on Spirit Stone and Spirit of the Elder Lizard don't stack with each other, you will only get the passive of one item.

Math

Squishy champions like carries or assassins have around 30-40 armor in the early game and around 80 armor (maximum ~130 with e.g. Zhonya's Hourglass) in the late game (with runes and masteries).
This build concentrates on reducing enemy champion armor to a minimum. You want to get their armor as low as possible because the damage increase is higher when you reduce the same amount of armor at a lower armor level. Example: Reducing the enemy armor by 10 from 110 to 100 gives you 2,6% more damage. Reducing the enemy armor by 10 from 10 to 0 gives you 9,1% more damage. Here is the link for the formula: LoL-wiki

The following calculations are without masteries (I will update that somewhen). You deal more damage with mastery % and flat armor penetration.

Let's say you have a Brutalizer and Ghostblade early and enemies have 50 armor at that stage of the game. You also have flat armor penetration runes that give you 19,2 armor penetration. You will reduce the enemy armor by a value of 49,2. That means that the enemy has only 0,8 armor left. That means you will deal 33% more damage.

Let's say you have purchased all items of the armor penetration build that will grant you 40 flat armor penetration and 35% armor penetration. Your runes still give you 19,2 flat armor penetration. That makes 59,2 flat armor penetration in total. At that point of the game your enemy might have purchased an armor item that gives 50 armor. He will also have 80 armor from runes and masteries (not auras anymore!). That makes 130 armor in total.
Percentage armor pen is applied before flat armor pen:
35% of 130 armor = 45,5 armor
130-45,5 = 84,5
84,5-59,2= 25,3 armor
This only counts for the first attack, every attack after it will deal more damage thanks to Black Cleavers % armor pen passive.
You reduce the enemy armor from 130 to 25,3. You will deal 36,5% more damage.

Armor penetration > AD

Early game you can't afford much AD but cheap items like The Brutalizer that also give AD and CDR. Your damage is dependent from the base damage of your abilities, you can increase it the most efficient way by getting a lot of armor penetration. Sometimes you will deal true damage to enemies or even reduce their armor to under 0 (you still deal more damage reducing an enemy from 0 to -20 armor). You don't want to lose damage in mid-late game, you should build Youmuu's Ghostblade and a second Brutalizer first for flat 30 armor pentetration, extra AD and CDR, also crit chance and a very useful passive that increases attackspeed and movementspeed. At this point you will burst squishies down very fast if you're not far behind (note that there are very hard counters to assassins like Riven that can disengage easily). After that you normally get Black Cleaver and Last Whisper or even a third Brutalizer (because the lower you reduce the enemy armor the higher is the % damage increase). At this point you have no problem with armor stackers anymore except when their armor is far over 200 (like Rammus) but you don't focus tanks anyway. You complete your build with defensive or high AD items.

This build can go wrong

Note that this build isn't perfect for any situation or when people know how to counter it.
The easiest way to be tanky against armor pen stackers is a mix of armor and HP. It's super dangerous to not build any armor against armor pen assassins since they will probably reduce your armor under 0 (if this happens your HP doesn't matter because your HP melts since they do over 100% of their actual damage). As a tank you want to get either a huge amount of armor so that the % armor penetration of the armor stacker doesn't reduce you to under 100 armor or You want to get around 150 armor and very much HP, very good items to get tanky are Randuin's Omen, Sunfire Aegis, Warmog's Armor/ Overlord's Bloodmail and especially Ninja Tabi are incredibly good early. As a carry you can build defensive items like Zhonya's Hourglass or Quicksilver Sash, those items are very good if the enemy assassin has only a short period of time to deal high damage (like Zed with his ultimate, Master Yi with his ultimate and Youmuu's active or even Wukong with his ultimate).