First off, "5e expects that at 4th level you will have an 18 or 20 in your prime requisite." is a conclusion I gained earlier, not a quote from the book.
The Proficiency bonus runs from +2 to +6. From 5th to 12th level it is +3 to +4. For the prime adventuring time, it seems, you can expect to get about +4 bonus for your level. An attribute of 16 to 18 gives a similar bonus at first level. At 4th level every class gets a +2 bonus to an attribute (or a feat that might give +1). This raises the attribute to 18 to 20. At a value of 20 the attribute gives a bonus of +5, which you won't gain from level until 13th level. Furthermore, 20 is the highest value an attribute can have under normal circumstances. Once your prime requisite reaches 20, you usually don't have much motivation to raise other attributes and can spend the ASIs for feats, unless you want to boost other secondary values, such as more hp from Constitution or boosting a particular saving throw.
Therefore, for the period of time that it seems most people enjoy playing their characters, you can have a base bonus of +6 to +8 to your roll in what you are good at. Magic items, tools, feats or features can add another +2 or double proficiency bonus, but for the most part that's your bonus for those levels and an attribute of 18+. At a bonus of +8, you have a 95% chance of making an Easy check (10), 70% chance of making a Medium check (15), and a 45% chance of making a Hard check (20). A typically heavily armored target will have an AC of 16 - 20, maybe.
Those odds will feel right at the table. It will be hard to remember a failed Easy check, except for that time your character was cursed and at disadvantage. A fight or situation against a Hard check will be memorable, and missing about half the time will make people seek out and feel rewarded by opportunities for advantage. Therefore, I think the game is designed to encourage the player to get as high a prime requisite as possible so that, right before prime adventuring time, at 4th level they choose the ASI and have an 18 or 20.
Now, reading through the PHB just now, I've discovered that 15 is the highest score you can get for an attribute through point buy. I thought it was 16. Even so, I think my point still stands.