The main purpose of filtration in a planted tank is to break down organic waste into simpler, less harmful substances by making use of natural microbial processes.
The most well known cycle is the conversion of toxic ammonia into less toxic nitrogenous compounds by bacteria and other microbes. The faster waste compounds are digested by bacteria action, the more efficient the conversion process, the better for livestock and the tank environment.
Plants and the substrate performs the same functions essentially - plants take in ammonia and nitrates and the microbial community in the substrate break down organic waste. In that sense, a well functioning matured planted tank can smoothly run without a filter, as long as plant uptake capacity is above the rate at which livestock produce waste.
In a new planted tanks, filtration is particularly impactful as there is usually a lot of plant debris and volatile organic compounds produced from transition stress. Plants may melt during transition stress, or old growth is replaced by new growth more suited to new parameters. These easily trigger algae blooms if organic waste is left undigested in the tank. The faster these organic compounds are broken down into simpler elements, the less of a trigger effect they have on algae. This is also why more frequent water changes is recommended in new setups for the first few weeks.
In matured planted tanks, with strong stable plant growth and a matured microbial ecosystem, this is less of an issue as the microbial population can digest the organic waste produced. However, it can take tanks many months to reach that stage.
Plants only take in ammonia/nitrates when they are growing well. Plants may not due well to a wide number of user errors - when they do not grow well, they contribute rather than take in nitrogenous waste.
Having a filter adds an additional layer of stability to a tank in terms of processing organic waste products should the plants not function well at any point in time. This backstop makes sure that you do not lose your livestock should your plants fail.
Most hobbyist planted tanks do go through periods of flux and instability, a filter acts as a counter balance to those periods.
Filters maintain water clarity by capturing fine particles. In biologically matured tanks, microbial bio-film binds fine suspended particles together. Filters speed up this process by introducing flow over a large surface area for bacteria colonization. If water clarity is poor - it can hint that the bacteria cycling process in a tank is not matured.
Most bacteria in the tank adhere to surfaces rather than free float in the water column - so a filter provides tremendous surface area for bacteria colonisation.
For many tanks a filter is used to provide water movement & flow. Water circulation in a planted tank distributes oxygen/carbon dioxide evenly throughout the tank and brings waste to filter elements.
Giving the importance of flow in a planted tank, it's important to have a pump for water movement if no filter is used.