You can find guides to what bullet/what twist elsewhere, but I will give you my 2 cents worth here and hope it makes sense.
One thing to visualize in the following discussion is that if you take a certain weight ball of Play-Doh and another heavier ball and roll them out to the diameter of a pencil, then the heavier ball of Play-Doh will be the longer one as well. If you shape a boattail and a pointy nose, the extra weight will have to go somewhere and so it will get longer again.
What the various manuals are referring to in .223 is the twist rate necessary to stabilize bullets in order to avoid them hitting the target sideways, if at all ("keyholing") as this picture by
Molon on AR15.com illustrates:
This is what happens when you fire the 75 grain A-MAX from a 16" barrel with a 1:9" twist.

The reason has everything to do with the length of the bullet. The longer the bullet, the faster the twist rate is required to stabilize the bullet in a nice stable spin (like a top). While this is generally related to bullet weight, once again there are exceptions if a bullet is very short or long for the weight. The main reason we go by bullet
weight is because that quantity is
known, and obviously a heavier bullet in a given caliber is longer. Boattail bullets are longer than flat-base bullets as well.
Originally the .223 was developed from the .222 Remington which uses a 1-14" twist to stabilize 50 and 55 grain bullets. So the very first Armalite AR15s had a 1-14" twist rate.
The long .223 55-gr tracer round was found to not stabilize properly in cold (lower velocity) arctic conditions with that twist so the .223/5.56 twist was changed to 1-12" and remained that way throughout the Vietnam war and into the early 1980s. This is also the standard twist for bolt-action sporters until recently.
When the European NATO allies decided to adopt the 5.56mm in the late '70s - early '80s, they decided they wanted a heavier bullet that would penetrate Soviet body armor at extended ranges better so they developed the SS-109 62-gr FMJ-BT bullet. This long bullet required a faster twist (at least 1-10"). When the U.S. adopted their SS-109 version, called the M855, they wanted sure stabilization and went with a fast 1-7" twist in the new M16 A2 rifle. While perhaps it is overkill (or overstabilization) for the lighter bullets, they will hit the target point-on with mil-spec accuracy. Sport shooter detractors felt this twist rate degraded accuracy of the lighter bullets so a compromise twist rate of 1-9" became popular in AR15s intended for civilian use. You could get decent accuracy from a wide range of bullet weights, omitting only the heaviest match bullets. On the other hand - National Match shooters used the fast-twist 1-7" to push new specialized ultra-long heavy BTHP match bullets of 75 to 80+ grains with their excellent ballistic coefficients to dominate 600 yard target shooting from the "mouse gun" and obsolete the .308 M14/M1A.
As alluded to above, the twist rate can have an effect on accuracy, with the ideal being a rate that stabilizes the bullet without adding any unnecessary rotational stress - once again there can be exceptions because of barrel smoothness and bullet quality so this is only a general guide, not gospel for every rifle:
With 1-12" twist you are most like to find happiness in the 40 (plastic tipped) to 55 grain range.
With 1-9", 50 to 69 grains with certain exceptions like 40-grain plastic-tipped bullets (
more on this next).
With 1-7" 55 boattails to 77 grains.
Another factor that plays into twist rate is bullet imperfection. The shorter a bullet is for its weight, the easier it is to stabilize, even if the bullet's not perfectly balanced. This is partly why early .22 centerfires had relatively gentle rifling twists, like 1-14" or 1-16" for the Hornet. Even a slightly imbalanced bullet will often shoot fairly well if not spun too rapidly. With the advent of modern bullets made on precision machinery, long range match shooting with the AR platform, and the change to the SS109/M855 bullet, twist rates have become much quicker for the .223 Rem.
The plastic-tip bullet revolution increased both ballistic coefficient and explosive expansion. The also became available in lighter than traditional weights, such as 40-grain. These bullets are now made on much more precision machinery as well so they are better balanced. This meant that lighter bullets shot accurately even in rifling twists that, according to older theory, were way too fast. It turned out that the big problem wan't rifling twist but the older bullets. Today it is common for 40-gr plastic-tipped bullets to shoot extremely well even in 1-9" twists.
To prove how well light plastic-tipped bullets like the V-Max or Ballistic Tip shoot in fast twists - a recent Handloader magazine test found that 40-gr plastic-tipped bullets (these are a bit longer BTW) shot the best of all lighter loads tested in a 1-8" Rock River Varmint AR15 and a 1-9" Savage bolt, with the accuracy edge going to the RR AR15. These were 1/2 MOA groups. However we need to remember that it was a smooth match-grade barrel with a long concentric high quality 40 grain bullet. Shorter 40 and 45 grain SP bullets intended for the Hornet shot very poorly but were plenty stable!
The
1-12" twist stabilizes spitzer bullets from under
40 grains to 60 grain flat-bases. An exception would be the short Speer 70-gr semi-spitzer that was made to stabilize in 1-12". Designed for 55-gr M193 FMJ-BT bullets.
The 1-10" and
1-9" twist stabilizes spitzer bullets from
40 grains to 70 grains. An exception is that
most 1-9" rifles will stabilize the shorter 75-gr Hornady BTHP, but sticking with the popular 68-69 grain BTHP match bullets as the heaviest in a 1-9" for all conditions is a sure thing for stabilization and accuracy.
The 1-8" and
1-7" twist stabilizes spitzer bullets from
40 to 80 grains. One issue with the longer versions of 75 and 80 grain bullets is that they are so long that they cannot be loaded to magazine length and must be single-loaded into the rifle. The illustrated 75-gr Hornady A-Max is one of that type. Standard military M855 is a 62-gr FMJ-BT and specialized military long range match is Mk262 77-gr BTHP match (Nosler or Sierra).
Like much of what I have found in my years of reloading and shooting - there are no absolutes (
well, we are quite sure that 75 A-Maxs won't shoot in a 1-9"!) but sticking in the general guidelines should give you an idea of a bullet weight to try. Other considerations are intended usage, length of bullet factoring in that boattails or plastic-tipped bullets are longer than standard flat-bases, and really long noses like the VLD or A-Max are really long bullets.
Another variable is the ALTITUDE that you are shooting at. Atmospheric pressure has a definite effect, primarily seen at different altitudes: Less twist is required to stabilize a bullet in the thin air of higher altitudes than at sea level. So stuff I can get away with at 6000 feet may not work at all at sea level.
In my shooting I have found that 55s shot better than 60 in my 1-12" bolt and that the plastic-tip Nosler Ballistic Tip BT 40-grain shot best of all against my own judgement.
In my 1-9" ARs both 55-gr Hornady SP (flat-base) and 55-gr Hornady FMJ-BT bullets shoot well. 60 V-Max not as well (could be powder used) and 68 and 69 grain BTHP match bullets shot MOA. I have never tried the 40s in these rifles.
I don't own a 1-7", but would if National Match-type shooting or use of Mk262 defense loads were intended.