It is indeed special in a sense that most aquarium setups include a lot of equipment, mainly a filter; the filter facilitates cultivation of nitrifying bacteria that are responsible for neutralization of harmful waste products - ammonia and nitrite. Without a filter, they would quickly build up to toxic levels and kill all animal life in the aquarium. These bacteria don't exclusively live in the filter medium/media, they generally stick to all surfaces like substrate, plants, etc. - but usually the filter media contribute to the majority of surface area suitable for bacteria, the filter is responsible for the great majority of their population in the tank and thus its operation is crucial for maintaining safe water parameters. It is not only a result of the filter providing a lot of surface area in its media, but also comes from a high rate of water flow through these media.
Aquarium presented in linked video is different - it uses a specialized approach, designed by Diana Walstad, known as Walstad method. I wouldn't say that it is particularly hard, but it is definitely different from the common approach and poses it's own unique challenges. It is designed to involve as little artificial equipment as possible. Its premise is to place an extremely nutrient-rich soil as a bottom layer of the substrate, then top the soil with a thick layer of fine sand. The upper sand layer serves as a buffer, isolating the soil from the water column and preventing the nutrients to be released to the water all-at-once. It is greatly slowing down the process of the nutrients leaching upwards into the water column or for the plant roots in the upper substrate layer to absorb. It extends the release of nutrients in time, which is a substitute for manual, periodic fertilization.
Many tanks don't require fertilization because usually the waste generated by inhabiting animals is enough to fertilize the plants; but the more planted the tank is, the more plant-related demand for nutrients it has. If one increases the plant density, at some point the amount of waste produced by the animals would be a limiting factor for plants and they would require usage of additional fertilization to fully thrive. Walstad method involves setting up a heavily planted tank - for that reason, it is remarkable for it to not rely on manual fertilization and still have a heavily planted aquarium with healthy and thriving plants.
At some point it is inevitable that the soil from down under becomes completely depleted of nutrients. But because the process of their leaching is extremely slow, it is expected for the soil to outlast the lifetime of the aquarium and go on for years; and even if it doesn't, one could always completely empty the aquarium, re-apply with the fresh soil and continue.
This method also relies on a large surface area of the fine and thick substrate to facilitate the cultivation of nitrifying bacteria that will contribute to the removal of nitrogen waste.
Walstad method, as mentioned earlier, incorporates setting up a heavily planted aquarium. Plants can directly absorb the mentioned nitrogen waste - but unless the aquarium is heavily-planted, their contribution to waste removal isn't enough and needs the support of a filter to prevent a constant buildup of waste. Walstad method is designed to mitigate this by heavy-planting the aquarium with fast-growing plants to provide a replacement for a filter - the plant density is so high that the aquarium's waste neutralization process doesn't rely on the presence of a filter anymore. Plants incorporate the absorbed waste in their biomass, so the faster-growing the plants are, the higher ratio of absorption is. One of the plants you could see on the video are duckweeds, which are these free-floating green things on the water surface. One plant consists just of a single leaf with a modified short stem, and they are remarkably fast-growing.
Many tanks do not need CO2 injection, but these are usually not heavily planted and don't involve especially demanding plants. But in this method's design, the decomposing material in the substrate and animals' respiration are the sources of enough CO2 for the densely planted aquarium, thus eliminating the need for using accessory equipment for CO2 injection.
One of the challenges of Walstad method is to never disturb the substrate to the point that the soil from down under mixes with the upper sand layer, which could happen for example during partial water changes. On the linked video you could see how the person manages this during partial water changes - a plastic container with small holes is placed on top of the water surface and then fresh water is slowly and carefully being added to the aquarium. It is also important to anchor the plants in substrate in a way that doesn't mix these two layers. Plants should be placed in a way that their roots are in the upper layer of the sand, but don't reach deep down to the nutrient-rich soil. Plant roots are meant to grow and reach this layer themselves, on their own.