Manufacturers typically list an age range that a typical child of that age could utilize most or all of the features of the toy in the way the manufacturer envisions the toy being played with. This may or may not match your personal considerations of what you find age-appropriate.
For example, when my son was 7 months old we purchased a plastic toy keyboard that was marketed for ages 18months+. At 7 months he was able to press on the keys and make them light up, but he was not able to understand the buttons to change the instrument sound and so forth. So while he wasn't necessarily using the toy to its full capacity, and may not have been playing music the way the manufacturer envisioned, he was still able to safely enjoy the toy.
Particularly when it comes to toys for babies the age range is determined by skills that a child of that age typically have. Like toys with buttons to press are usually labeled for 6+ months (or more) because babies younger than 6 months typically do not have the dexterity to press buttons. Or a ride-on toy requires the ability to both sit up, and climb on the toy.
If a child is given a toy that is significantly too hard, they may find the toy boring or frustrating. They may not have the manual dexterity necessary to operate the toy, they may find putting puzzles together too challenging and don't know what to do with the pieces and so on. Toys that are too easy, likewise will not hold a child's interest--give a child over the age of one a toy labeled for a 3 month old and they will probably be bored of it within minutes, and not find it exciting, and prefer more age-appropriate toys
There are different ways manufacturers can determine an appropriate age range.
One would be though focus groups or other forms of product testing, where they have children of different ages play with the toys and they see which ages enjoy the toy and use it appropriately.
Another would be by considering the skills necessary to operate the toy and looking up what age, developmentally, has that skill. A toy that requires reading and following specific instructions would require a child old enough to have a sufficient reading level to understand the directions and be able to follow them. The directions a 5 year old can follow must be much simpler than the directions an 8 year old will follow.
In some cases the age range may have been set by a marketing department before the toy was created, based on the demographics of their target audience. Or they might compare the age range of a competitor's product and choose to market theirs similarly.
In many cases, the age range is arbitrarily selected to simplify product testing.
From a well-written article on Slate.com about how toys are safety tested:
Since no toy manufacturer wants to get stuck making an embarrassing recall, most companies have their products evaluated by independent testing labs before they go on sale. [...] Many stores won't buy a product unless they're satisfied that it meets these standards.
What kinds of tests does a toy have to pass? The generally accepted standards, testing methods, and labeling requirements are laid out in an ASTM document called the Standard Consumer Safety Specification for Toy Safety. Every kind of toy must generally be clean and "free from infestation," and it can't have sharp edges or exposed bolts. If the toy is meant for children under 3, it can't have any "small parts"—defined as pieces that can fit inside a 1-inch by 2.25-inch cylinder.
So, basically, it is not uncommon for toys to be labeled as 3+, simple because a company does feel it is justified or cost-effective to spend the extra money to have their toy subjected to additional testing to verify whether or not it has any parts that are choking hazards and the like, if they expect their primary users of the toy are not babies. I have even seen fuzzy baby-rattles in a college bookstore say they are for ages 4+ on their tag--which would appear to be more for lawsuit-prevention purposes than based on what age would use and enjoy the toy.
Generally, I would say that toys for ages less than 3, the age range is based on the typical developmental abilities at a particular age. However, obviously, some children walk and talk younger than others and have varying skill levels, and in some cases, a child may simply never like a toy, age appropriate or not--my 18 month old son, for example, is quite fascinated with the ball popper you linked to, and studying how the balls defy gravity. Some of it is personality. The toy age range suggestions may help a parent decide that one type of toy to select, but they are not absolute, and many children are ready for toys more "challenging" than the package lists the ages for sooner. But sometimes a child will not be able to immediately fully enjoy a toy if given too soon. At 12 months, my son could not solve board puzzles, but at 10 months, he can. But that doesn't mean we waited until 18 months to obtain one.