Valk's analysis is more accurate for how to value the item active, though the usage considerations are still worth noting.
While the average value of the passive is gonna be based on the item's uptime, which maxes at 6 seconds out of 45, the value of the active gets skewed by burst scenarios, or using it in appropriate windows to get more value out of it.
So if you get 9 seconds of wailing on a turret due to the conditions on the map, you get that active uptime for 2/3 of the relevant time and get 1193 average gold value for that interaction. If you didn't need the movespeed for dodging during that time, then it plays more like 667 gold value for that time period.
The average max value is still going to approach but not actually reach 238 gold due to not popping it on cooldown.
And if you're caught out while it's on cooldown, you get it for 0 seconds of that interaction, and get 0 gold value during that.
How you use the item is extremely relevant to the value of it, because if you're consistently layering that 6 seconds over short interactions when they matter most, you get a lot of value out of it. Obviously very good players are going to tend to maximize the use of it.
The active isn't why it's bought though, it's that the general stats are nice. Penetration is a geometric scalar like other auto attack related stats. If the enemy has like, 100 armor, then you're getting 11% more damage out of all of your physical damage due to the 20 ArPen. (100+100)/(100+100-20) = 1.111ish. If they have 60 armor, its (100+60)/(100+60-20) = 1.14ish, for 14% more. That'll range from 16-13% on turrets too, which crit doesn't work on, due to turrets now ranging from 40 to 70 armor throughout the game.
An I.E. costs more, and it's bonus only adds to autoattacks. The average value it adds being 30% to autos only, and is very erratic early. (0.2 * 1.5)= 0.3, for 30% more average auto damage. That'll pop to a harder spike with a zeal item, but when early trades are dominated by ability use, and the ArPen still scales multiplicatively with the crit you build later, Youmuu's is obviously pretty efficient. It's cheaper, so you spike earlier, and you're more consistent. It's a good lane winner, and has a friendly build path.
The comparison's still wonky since B.F. into Zeal item into I.E.'s been more effective than straight I.E. for a while, but the point remains. Youmuu's active is gravy. Armor rushers are the only ones that'd make the Youmuu sub-par, and they don't make up that much of the enemy team. And I.E.'s reliance on a 2-item spike is rough with the early game focus.
This mostly applies to the ADCS, but AD casters mesh even more naturally with the ability scaling Youmuu's gives. Plus CDR. Because CDR.