Others may have vested interests in maintaining their beliefs. They may benefit personally from limitations on others' freedoms. Or they may enjoy the power or prestige of making and enforcing decisions for other people. They may even truly believe others are incapable of functioning without guidance from a higher authority. If liberty truly is the best way to arrange human society, then following the principles of freedom should, in fact, result in the best outcomes.
Results are important.
In this type of discussion, it is important to be aware of the facts, statistics, and history of the particular topic you are arguing. If you seek to convince someone that laissez faire capitalism is the best economic system, you need to be able to present information on such things as comparative lifestyles, productivity, technological innovation, and wealth creation. You should be conversant not only with the present situation but be able to place it within a wider historical context. The more you engage in argumentation, the more you will learn what points will likely be brought against you. If others claim the Industrial Revolution led to the exploitation of children, you need to be aware that capitalism inherited massive poverty.