How does the statement of purpose of students help professors to write letters of recommendation for graduate school application in U.S.?
The statement of purpose is helpful to me in writing a letter of recommendation for a student because I can say things in the letter that relate to the specific plans that the student has beyond the more general plan of entering a graduate program.  

For example, a student might ask me to write a letter of recommendation for admission to a PhD program in applied mathematics at University X.  Knowing that the student plans to specialize in a subfield (say computational optimization) gives me the chance to write into the letter that the student is well prepared for graduate study in this area because of (whatever factors might be relevant, including specific courses taken, programming languages the student knows, etc.)  Those same experiences might not be relevant if the student wants to work in a different subfield.

Another reason for me to want to see the student's statement of purpose is so that I can provide helpful advice to the student.  Many students are naive and prepare draft statements that won't help and might actually hurt their chances of admission.  If I've read the draft statement I can suggest changes.  If it's really bad, then I might decide that I'm unwilling to write a recommendation letter for the student because it would reflect badly on me.  

Ultimately it really doesn't matter why the recommender wants to see your statement- you're not in any position to demand a letter of recommendation, so you should provide whatever information the requestor asks you for.