"把(bǎ)" structure and "被(bèi)" structure are both very significant structures in Chinese.They represent to "active voice" and "passive voice" in English."把" structure is basically formed by the following parts:subject(A)+把+object(B)+action.It means "A does .... to B".The structure emphasize on consequence or method of an action.In modern Chinese,object is usually placed behind verb,and the usage of "把" structure can shift the object in front of verb,therefore can emphasize the result to the object caused by the verb.E.g.,我把苹果吃了(wǒ bǎ píng guǒ chī le,I ate the apple.).It implies that what has been eaten is an apple.The usage of "把" structure has three necessary requirements below:First,the verb must be transitive;Second,there is usually an additional part in front of or after the verb that states the consequence or method of the verb;Third,the object of "把" is usually certain,which means the speaker need to have a certain object to point at.This structure is quite commonly used in imperative sentences,such as the example you gave. Another structure "被" is also very important as "passive voice",but I'll just leave it alone here for now. As for that "来(lái)",it is a very special usage of this character.It is used as a directional complement that states the trend of a verb or a verb phrase.The whole sentence is still completely correct without "来",but using this character can make the sentence sound more natural and fluent.It is very difficult for beginners to understand the usage of "来" here,so you can just ignore it for now.

Hawaii has well-presented most of the gist. However here is something from another perspective: 请把书打开来 （qing ba shu da kai lai） First of all, you may notice 请（qing）in the beginning, which suggests this is a imperative sentence, by which someone ask you to do something. 请（你）把书打开（qing ni ba shu da kai lai） See? In fact there is an omitted "you" right after the word "please". Then you can understand why there is a "ba" to state the "active voice". Second of all: I don't agree with Hawaii about the usage of "来Lai" putting 来/去 (they mean come/go when they are used as verb) in the end of the sentence suggest a transition in states. In this case, it means the book should be open and it's redundant. The whole sentence is less natural after you use 来 in the end. Delete it and then you have what us Chinese usually says. By the way, I am a Chinese, born and raise in China.