As already mentioned, plants literally eat up (in a way) the toxic stuff found in the water that the betta creates. This makes the plants happy and the fish happy. Plants also do create oxygen, however, this not not benefit betta fish in particular, this would be helpful for fish that do not posses a labyrinth lung.
While beneficial bacteria does play a part in the process of cycling a tank, there are really two ways to cycle a tank, one is with movement of water in order to support a high enough level of beneficial bacteria (BB) that turns ammonia to nitrites and then nitrites to nitrates. Thus rendering the water not toxic to the fish as waste is broken down. The second way is through plants using the ammonia/nitrites in order to grow. So while some BB might be present in the water, it is not the BB that breaks down and detoxifies the ammonia/fish waste, but instead the plants absorbing it and growing from it. Plants also absorb CO2 and create oxygen from it (basically the opposite effect from our breathing). This is crucial for fish that do not breathe from the surface like betta fish do, especially if there is not a filter or other method to oxygenate the water.
For most plants, anything that oxygenates the water, such as a filter, or air stone, and creates movement will make it more difficult for the plant to grow. That is, filters and air stones remove CO2 from the water, which is what the plants need to breathe. While these items are absolutely crucial for keeping some species of fish alive when plants are not present, it will stunt the growth of plants. In order to increase the growth of plants there are two popular options: the first and cheapest/easiest is to use a CO2 replacement. This is often a liquid that is added to a tank on a regular basis. However, some types of mosses (Christmas, java) and algae (such as the Marimo moss ball type) will die or brown when this is added to the tank. The second option, and the one the professional aquascapers often use is to inject CO2 into the tank. This can be costly to set up and maintain, but the reward is plants that can grow inches a day, turn vibrant colors if their species allows (such as red or purple), or even difficult species will grow well. All plants benefit from this, even the aforementioned ones that might die with the presence of a liquid CO2 replacement. There are ways to make your own CO2 system at home for much cheaper than the expensive injection systems that are available as well, but your results may vary.
Remember that the tank your fish lives in is his or her's whole world. Ask yourself this question, and answer it honestly: would you choose to live there?