An honest question. I just don't see how this is guaranteed a better route as opposed to evenly distributing your ability points as you level up :3
Maybe I'm just dumb, but some explanations would be greatly appreciated :D
Basically its because not all abilities have the same worth to the champion. Let's take Jinx for example because she is easy show. Her Q is most valuable on her kit because of the attack speed it gives for taking turrets and the range on her rockets. You can out range a Caitlyn by level 7 by maxing it first. Her w is a strong skill but Jinx will attack so dreadfully slow and at such a short range if you don't max Q first. Once Q is maxed W is next because its her damage spell. She is an adc; dmg is what she wants. E is a one point wonder skill. You don't use it for damage so maxing it early for that is pointless. CC is what its used for and that doesn't scale with points so it doesn't need points till late. And ultimates by their very nature should be leveled whenever available.
Most champs follow a pattern like this where one ability is priority in the kit for the champ to work/deal the most damage (maybe a skill increases 25 dmg a point vs 20 on another) and poses one spell that is a one point wonder (like a cc that doesn't have duration scaling). Some champs max skills first based on situation. Some Riven's for example max shield first (rather than Q) against losing lanes because she deals damage with her passive anyway and needs the defense.
There are multiple reasons. First, champions usually have one ability which has much more versatility and general use than the other ones, and they have abilities which function very well at only one point. Take Graves for example. Buckshot (his Q) is the ability he'll get the most use out of, and maxing it gives damage, waveclear, ranged poke, etc. It won't do any of that effectively without levels. Meanwhile, his W vision denial functions perfectly fine with only one point, and his E dash will give him mobility regardless of whether he has one point or 5.
On top of that, having one ability leveled lets you fight more efficiently. Say I'm playing Lux. If I have E maxed, I can throw that out for poke and save my Q. This not only allows me to poke better, but also lets me keep my Q for CC if I need it. It also improves my mana efficiency by only using one ability instead of 2 for the same amount of damage. The general effect is that having points in E is always better than splitting points between E and Q.
Do you have any specific champions you would evenly distribute your skill points on? I can explain for any given champion why there is an ability they would want to max.
Basically some skills benefit alot more from being maxed out as soon as possible. I'll use 
as an example.
His E is his most important source of damage, and as a mage. You generally want to deal lots of damage.
While at the same time, Early game your Q is another source of damage, it doesn't scale nearly as hard as your E does. But since its still another source of damage, you max it second to maximize your dps.
Then you have your W, which is 100% utility. Meaning you can generally put 1 point into it early game and be fine, since your not a support champion. And more of a dps mage who wants all the damage you can get. Since the only thing you get from additional points in your W is a lower cooldown and better slow. The lowered cooldown generally isn't noticeable until late game.
In short, every champion wants to get their most important spell maxed as soon as possible. While utility spells can get away with just one point in it for awhile.
Because if your fed you want to have oppressive damage in one skill so that you can 100-0 people.
It's the same reason that you skip defensive items and build mostly offensive items if you get fed as a bruiser. Why be a well-rounded fighter when you can 3-shot people and end the game well before the enemy team has a chance to get caught up?
For every champion, some abilities just level better. It's that simple. You already got a good long explanation, so I'll just post a short, general agreement.
Generally, because one ability will grant greater benefits per rank than another.
I know you've already got a bunch of examples and your question is answered, but my favorite way to put it is with Talon's kit. His W provides the largest damage bonus, +50 base damage per rank as well as increasing the slow's effectiveness. After that, his Q gains +40 base damage per rank and a lower base CD. Finally, his E increases the damage amp by 3% and reduces the base cooldown (and percent damage amps scale better with higher numbers).
Now, sometimes, it won't be as clear-cut which one is better (Udyr). But, basically, you're maxing the ability that benefits what you want upgraded most first, and keeping the others at lower ranks because you need the ability's function rather than its full strength.
The one you max first is often the ability you use the most. Let's look at this guy: 
Consider these two possibilities for your first 8 skill points.
Q-E-W-Q-Q-R-Q-E or Q-E-W-Q-E-R-Q-Q
Practically the same, except that you take a second point in E at level 5 instead of level 8, so that your skills aren't so lopsided. The difference is that during levels 5, 6, and 7, your Q is 1 level below what it would normally be, and your E is 1 level above what it would normally be.
About how long does it take to get from level 5 to level 8? About how many times do you think you use Q during that time? Like 20 or 25? And about how many times do you use E during that time? Like 5 or 6? Is it worth giving up 40 damage (total on 2 hits) on all 25 of those Qs for being one rank behind, just to get an extra 30 damage on all 6 of those Es for being one rank ahead?
Now apply that same formula to the rest of the game, and to a truly even skill distribution. You'll be slinging like 100 weak-ass Qs from levels 5 to 15 just so that your 20 Ws and 20 Es will be stronger.
When laning 
, I usually put two points in his W first, then max his Q. The reason behind this is that Cho's W is better for harass early game, but the 1.0 scaling on his Q is better for mid-late game damage and CC.
