When looking at a pure form of laissez-faire capitalism, economic and business decisions are not to be influenced by the government. Property and means of production are all controlled privately as well. Capitalism is based around the idea of supply and demand, which helps to determine what will be produced, how much is produced, and what the cost will be. A greater demand and a lower supply means prices will go higher, while a lower demand and greater supply will mean prices will go lower. Capitalism also relies on desires of profit or gain to function, as profit determines a business owner's income.
As income is related to profit for business owners, that creates a system of competition between business owners. Let's say, for example, that a car dealership is the only car dealership in a very isolated town. They control the supply (cars available for purchase) in this market. It would be hard for the people in town to go to any other dealership. Therefore, the dealership does not have to worry about competing with other dealerships' offers and holding special sales to entice customers. Basically, if you want a car in this town, you need to get it from this dealership, otherwise you have to find some other form of transportation. Now while they can't charge whatever price they want, because people simply won't be able to afford a car, they can charge a relatively high price because there are not other easily available options.
Now imagine that a new dealership opens up. Suddenly there is no longer control of the market. While the town's population remains the same, and thus demand for cars remains relatively the same, there is a big increase in supply. This means that the original dealership stands to lose money. They now have to make sure their prices are competitive with the new dealership, which likely means lowering them and thus eating away at some profits. Also, they may lose some customers to the new competition, which further eats at their profit. The end result is that the owner of the dealership is likely not making as much money as they previously would have.
When examining this situation, we can see that capitalism, which is fueled by profit, can lead to a situation where there is greater personal interest. The owner of the original dealership would likely be happy to see the new dealership fail, even though it is beneficial for car buyers because the price is lower. I think it is important to consider this is a very basic example, but I do think that it highlights how the capitalist system does breed a heightened sense of personal interest.