They're not saying that there were no problems between people (we'd identify as) of different races before the concept of race arose; of course there were. "Racism" vs. "Harmony" is a false dichotomy. You don't have to really get too far into the history: race as a rigid category with scientific pretensions is more recent and specific to Western imperialism than ethnic/tribal/regional/linguistic lines of conflict/oppression more generally.

Some sources:

Robert Wald Sussman The Myth of Race: The Troubling Persistence of an Unscientific Idea

Thomas F. Gossett Race: The History of an Idea in America

Edit: original first line not meant to sound snarky, sorry.

The racism we think of today is partly based on (pseudo-)science. An offshoot of taxonomy, to be precise: the idea that some peoples were innately inferior and deserved to be ruled. This more or less displaced the earlier, Christian idea that people were equally God's creatures. In the Middle Ages, people were prejudiced against people of other religions, but not so much against people of other races. Note, for example, that Malory's depiction of the Round Table includes a "Moorish" (black or middle-eastern) knight, Sir Palomides, without making a big fuss over it.

You might be interested in some previous threads about about the origins of modern racism and racism before the Renaissance:

Racism and European Imperialism

• Europe had normal diplomatic relations with non-white nations before turning explosively racist to justify their actions against all others. What happened to cause this shift? - 22 comments, over 1 year old.
• A flaired commenter in this thread provides a multi-comment long overview of the "flowering" of racism and scientific racism in the 18th and 19th centuries at the height of European imperialism.
• What are some of the more notable (scientific) critiques of scientific racism - especially in the 19th century? - 3 comments, over 1 year old.
• A flaired user's description of the opposition to scientific racism also serves to highlight the common racism of the day and its origins.
• When did the concept of Race begin to emerge? - 25 comments, over 1 year old.
• There are a few summaries of "milestones" of the development of racism from the Age of Exploration to Age of Imperialism with a related article linked in the topmost comment.

Racism and Slavery

• Did Racism Cause the Enslavement of Africans? - 2 comments, over 2 months old.
• The commenters here summarize the matter without going into detail.
• Do you think racism was the result of slavery? or whether africans were enslaved because of racism? - 7 comments, over 8 months old.
• The commenters here go into detail on the genesis of racism in the US from the Thirteen Colonies to the Antebellum United States.
• Why did Europeans enslave Africans instead of other Europeans? - 17 comments, over 2 months old.
• The commenters here go into the difficulties involved in enslaving people in Europe to compare and contrast with the methods used in Africa as well as going into developing attitudes towards race in the Early Modern era.
• "I don't think that ancient slavery is really comparable to the chattel slavery that we saw in the Americas." How did ancient slavery differ from the atlantic slave trade? - 91 comments, over 6 months old.
• This thread is mostly about Roman slavery including how slaves were acquired, what they were used for, and what ethnicity/race they were and thus allowing comparison with the situations presented in the threads above.
• How true is the claim that "racism created race", as opposed to the other way around? - 9 comments, over 6 months old.
• This thread brings up and summarizes the a book with a contrary claim to the traditional narrative that modern racism primarily derives from the Early Modern era.

Racism in the Middle Ages

• How accurate are the assumptions made by the blog medievalpoc concerning nonwhite peoples in medieval Europe? - 7 comments, over 11 months old.
• A flaired user for Medieval Western Europe gets into the complexity of actually piecing together instances of racism from the Medieval era that fit modern definitions of it from the written evidence available.
• What role did race play in European Medieval society? Did the concept of race develop over centuries? - 21 comments, over 1 year old.
• The commenters in this thread discuss certain aspects of racism present in the Middle Ages in context with the highly prevalent religious discrimination of the period.
• What did pre-modern racism look like? - 14 comments, over 2 years old.
• Ignoring the problematic Alexander the Great quote at the top, a commenter towards the middle of the thread discusses anti-Semitism in the Medieval German states.
• How prevalent was racism in medieval Europe? - 3 comments, over 1 year old.
• A commenter discusses anti-Semitism and xenophobia in Medieval England.
• How did medieval Europeans reconcile their anti-semetism with their Christianity? - 9 comments, over 6 months old.
• The commenters discuss the links between anti-Semitism in Europe and the development of European Christian identity.

Racism in Antiquity

• I've often heard it said that the ancient Romans were so culturally and ethnically non-homogenous that "racism" as we now understand it did not exist for them. Is this really true? - 137 comments, over 2 years old.
• This thread dives headlong into the matter, inciting debates over whether the ethnic stereotypes common in the ancient world are of the same kind as modern racism or not while supplying plentiful evidence and analysis for both sides.
• Was there much racism in the Roman Empire directed at people from other regions? - 47 comments, over 1 year old.
• The commenters in this thread differentiate between the xenophobia present the ancient world from the modern understanding of racism.
• I know the Romans had no conception of race, but did they view Black Africans any differently than they did the Persians, Libyans, or Celts? - 80 comments, over 11 months old.
• This thread goes into the way Rome interacted with peoples that would be classified in the modern day as being of different races.
• Racism in the ancient world? - 37 comments, over 1 year old.
• This thread pulls examples from all across the world and its history along with copious amounts of sources for further reading.
• Did the Romans have stereotypes about the behavior of people from particular tribes or geographic areas? - 9 comments, over 11 months old.
• Several commenters detail the most common stereotypes that Romans held of others at the borders of their empire and beyond.

"Racism" as in thinking of or at least feeling that other ethnic groups are inferior or bad is part of human nature, culture has little to do with that.

As for the classification thing, that got big with colonialism and after evolutionism, with scientists attempting to apply taxonomy (classifying animals) to humans.

Classifying people by race (with those who came up with the idea of "race" on top, naturally) gives an excellent "scientific" justification for people's hate and for exploiting colonies in Africa and elsewhere, which explains why it was so eagerly accepted while exactly that was happening. That is what your book is referring to.

There was probably never "racial harmony" anywhere for very long, but the pseudoscientific aspect and the nice little color-coding (as opposed to just referring to the others as "them") is, indeed, mainly due to colonialism.

Scientific racism (modern racism) in it's "enlightened" form were European "inventions". Hating people for being different not so much. This is as old as tribal living.