Why did the changes that the civil rights movement brought about matter?

The civil rights movement of the 1950s and 1960s mattered because it finally led to the passage of legislation at the federal level that invalidated racist state laws. During this era, the country finally moved away from apartheid (separation of the races) and towards integration.
The Civil Rights Act of 1964, for example, made segregation in public places illegal. No longer could black people be forced to use separate bathrooms, sit only in a segregated part of a movie theater, such as the balcony, or be kept off public beaches and out of public swimming pools. No longer would restaurants or lunch counters be only for whites. This has been a lasting change: today it would be incredible to think of putting up different water fountains for whites and blacks or refusing to serve black people in a restaurant.
The Civil Rights Act also banned employment discrimination based on race.
The civil rights movement built sympathy for the plight of black people through its persistent non-violent acts based on Gandhi's earlier political movement to free India of English rule. Other notable civil rights achievements include the Fair Housing Act of 1968, which made it illegal for real estate agents or local jurisdictions to ban black people from living in certain areas, and the Supreme Court decision that invalidated the laws in many states forbidding marriage between black and white people.
The civil rights movement was remarkably successful, but work is still left to be done to eradicate racism in the United States.