prepositions - She comes from/of a rich family

She is very arrogant because she comes of/from a rich family.

of is the correct preposition here. But how does using from here makes it grammatically wrong?

Answer

Neither comes from nor comes of is wrong.

The more commonly used preposition in Modern English (for this pattern) is from.
Using of makes it sound archaic.
Also note that with of, the verb be was more usual¹ (e.g., She is of a noble family).

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This Google Ngram chart shows the results of "comes/came of/from a noble/rich family":

As you can see in the chart, came of a noble family was once a common phrase. It reached its peak in about 1910. The recent trend of came/comes from is obvious, particularly in the last few decades. So your book is not wrong; it just teaches you the common usage used in another century.

¹Also note that although came of a noble family was once common enough, even more commonly used phrase was was/is of a noble family, as clearly shown in this chart:

I added is/was of a noble family to the first chart, and it dwarfed all other patterns at once. Also obvious in the chart is the fading of the trend of "[be] of a noble family".

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To wrap up this answer, (... because) she is/comes of a rich/noble family will sound archaic. The more usual phrasing in Modern English would be: (... because) she comes from a rich family.