What this statement refers to is something that Alexis de Tocqueville called the "tyranny of the majority." Although the idea was not new with him (it is a major idea in Federalist 10) we use his name for it. The basic idea here is that, if you give the majority unlimited powers, they can at times use those powers to oppress minorities. This is why we have a system with checks and balances and a limited government with written limits on its power (the Bill of Rights, for example).
You can think of examples from our history where majority rule has abused the rights of minorities. There was the segregation era when the majority was allowed to deny blacks their rights. There was the time of McCarthyism when the majority denied the rights of speech and association to those who held the wrong beliefs. There are other examples in our history.
So, the basic idea of this quote is that the majority can use its power to abuse minorities if there are no checks on what it can do. In order to combat this tendency, our system has constitutional guarantees of rights and a judicial system that has the power to override majority opinion when the majority is violating those guarantees. (Of course, as mentioned above, even this doesn't always work to protect us from the tyranny of the majority.)