Thus they are compelled to cut all connection between their statements and experience; and precisely by this procedure they deprive them of any sense
In deprive them of any sense the word "them" refers to "their statements" in the preceding phrase. "Deprive" means "to take away from" or "to withhold from". "Sense" in this context means "meaning". So the sentence with deprive them of any sense is saying that their statements do not make sense because they are separated from experience.