Such a headline is often used to bait newbies to the proposition of having a planted tank without sophisticated equipment or the need for dosing. By using soil and easy to grow rooted plants, it is very possible to grow simple planted aquariums over many months without fertilization as the plants can draw what they need from the soil. Fish waste can also provide a good amount of essential nutrients like nitrogen and phosphates.
However, all soils deplete over time, most soils will have enough stored iron for many months, but water soluble nutrients like potassium deplete quickly. Going the "No ferts, no dosing" route is basically a bet that your tap water has a good supply of essential minerals that make up for what fish waste cannot provide. Fish waste is not a wholesome source of nutrients- looks good in theory, works terribly in practice despite the claims of some books. Fish waste provides good amount of nitrogen and phosphates, but rarely adequate amounts of potassium & magnesium, and no fish I know excretes chelated iron as waste.
So, depending on your tap water, it may be very doable in certain areas, for a period of time, especially in a combination of good soil and easy plants. Rarely does this result in optimal growth though.
The few examples that do make it are paraded as the norm (classic survivor-ship bias), but its not a dependable method for most folks. The typical 'no fert, no dosing' tanks do terribly, as plants quickly starve of essential nutrients and deteriorate - this then leads to algae. This is why lushly grown planted tanks are not the default form of planted tanks in the world.
Simple dosing, even once or twice a week, goes a long way to provide a stable baseline for growth. A larger variety of plants can be grow in optimal condition than just the easiest of species. Given the cheap price of liquid fertilizers, there is really no reason not to dose minerals to make up for what the tap water/fish waste lack.
Many natural water bodies have elevated CO2 levels due to organic decomposition and sequestering of CO2 in underground reservoirs. Spring water typically has much higher CO2 levels than our tanks and such environments that support thriving plant growth typically have elevated levels of CO2 in the ranges of (10 - 40+ ppm). This is much higher than the 2-3ppm that standard non CO2-injected fish tanks will have (in equilibrium with atmospheric CO2 levels given gas pressure laws). To this end, CO2 injection brings CO2 levels closer to many areas in nature. Read more here on CO2 levels in nature.
"I’ve measured dozens of rivers and lakes at 8-30 ppm of CO2 natural gas levels. They are spring fed, think Perrier bubbling CO2 carbonated spring water...all spring water is enriched with CO2." - Tom barr