In very broad terms, the CR of a creature is based on three factors:
Numbers. Attack bonuses and AC, save DCs and saving throws, etc. etc.
Requirements. Magic weapons for overcoming DR/magic. Flight, or ways of dealing with others having it. Teleporting, or again ways of dealing with it. Protection from various status effects (e.g. those things that protection from evil, freedom of movement, death ward, mind blank, and so on block).
“Puzzles.” Anything that requires a very-specific trick: a particular type of energy damage to overcome regeneration, ghost touch or similar for incorporeal creatures, and so on.
In all cases, it is very important to remember that CR is notoriously unreliable. Even if you run everything by-the-book and have an exceptionally generic party, CR cannot be relied on; any given CR has a wide power range that overlaps considerably with the CRs around it. That is, you can always find relatively-powerful CR x creatures, and relatively weak CR x+1 creatures, such that the lower-CR creatures are actually more dangerous. So just keep that in mind: any statement about CR has to be taken with a huge grain of salt, because the system itself is so unreliable.
Creatures whose CR is based primarily on Numbers
These are creatures who don’t have much in the way of particular tricks, even fairly standard ones like flight or DR/magic. If they have them, they have things that have been common for many, many levels, to the point where you would never have made it this far if you couldn’t handle them.
Instead, they’re just tough. This doesn’t have to be physically tough; they could be particularly mobile and have nasty spells rather than a ton of HP and a giant weapon, but even in the “magical” case it’s not really all that “tricky.” After all, on some level the difference between an archer and a blaster is somewhat academic.
This is pure math. Magic items are often very important to the math, but until mid-to-high levels, the differences aren’t make-or-break. mxyzplk’s answer covers this well. In my own games, 7 and 12 look a bit high, but I tend to play with people who expect a certain level of optimization, so we expect numbers to grow faster.
Creatures whose CR is based primarily on Requirements
These are creatures who are on the cutting edge. CR 2-3 monsters with DR/magic, to really encourage you to get a magic weapon. CR 7-9 monsters who fly and attack from range, and make it very, very difficult to engage without flight of your own. CR 9-13 monsters who teleport frequently and are extremely difficult to keep up with if you can’t too.
Their numbers may or may not be up to par, but if they’re not, they’re close, and if they do, they’re not anything special. They are competent and they have powerful but standard tricks.
For these, the mundane classes are in serious trouble. In almost all cases, the “requirements” I mention are magical in nature; if you aren’t a magical class, then magical items was going to be your answer to these creatures. And these are literally the creatures who exist to punish you for not meeting these requirements, for skimping on something the game considers standard fare at your level.
For example, in a typical game, a hypothetical CR 5 monster can have DR 20/magic and most people won’t actually care: DR 20 is massive for 5th level (more than you’ll actually see), but almost everyone who wants to attack with a weapon is going to have magic weapons. Even the monk does. In fact, even CR 3-4 monsters can have a pretty good chunk of it, and a lot of people are prepared for that.
In your game, such a monster might literally be impossible to kill, if they haven’t gotten magic weapons. Even the very-typical DR 5/magic can turn a tough fight into suicide, and a cakewalk into a tightrope walk.
Puzzle monsters are things that really can only be killed one particular way. Trolls, hydras, swarms, ghosts, that kind of thing. Very often, puzzle monsters are quite weak for their CR – if you have the trick.
The problem is that the tricks that you need are almost always supplied by magic. This is the “requirements” problem, but taken to an extreme and showing up far earlier.
Moreover, because magic is massively more versatile – that is, spellcasters get far more spells than warriors get feats and so on, and spells are not restricted by “verisimilitude” – the spellcasters can have far more tricks available simultaneously. This is especially important for puzzle monsters, because they need something very specific: since mundane classes get very few options, they need to take only those things that are very broadly applicable. Spellcasters actually have the flexibility to prepare that one thing you’d never normally need; mundane characters, generally speaking, do not.
You need to be able to judge why a given monster has the CR that it does (also, if it ever deserved that CR in the first place; many deserved CRs higher or lower) in order to do this.
For number-based monsters, the problem isn’t too bad, or at least not until mid-to-high levels, and the players can always ease their way in after grabbing a couple of levels if they need to.
For requirement-based monsters, the problem can be a lot worse, but at the least it should be possible to both anticipate problems (you can see that they lack something the monster expects them to have), and quite possibly to telegraph dangers to the players (e.g. rumors like “The creatures of the Sharp Rocks are naturally protected from physical injury; only magic weapons really work on them.”). But the key is that you have recognize such monsters!
For puzzle monsters, honestly the situation is quite similar to requirement-based monsters, just more extreme. Further, these are an excellent opportunity to exercise some creativity, and set up such fights where player cleverness can replace magic items. Maybe trolls live in a eucalyptus forest; maybe the ranger knows that those are terribly prone to burning quite vigorously. Maybe you don’t need a flaming longsword so much as just a simple torch. Maybe ghosts are haunting a temple, and the priests can bless your weapons with ghost touch – but they have to be physically present and maintain concentration to do so. Or whatever. But definitely recognize these monsters, and be careful using them when you do.