How is one supposed to "know" whether or not the Book of Mormon is true? When one is introduced to the Book of Mormon, he is oftentimes asked to pray to find out whether or not the Book of Mormon is true.

According to LDS theology, why is a person simply asked to pray about it, rather than looking at historical, archaeological, prophetical, and other fields of evidence that attest to the authenticity and divine authorship of the Bible?  Also, how exactly is one expected to know whether or not God has confirmed that the Book of Mormon is true?  How is God expected to answer the person's prayer?
To understand the answer, it's important to understand the actual invitation.  It can be found in the last chapter of the Book of Mormon:


  Moroni 10: 4-5
  
  4 And when ye shall receive these things, I would exhort you that ye
  would ask God, the Eternal Father, in the name of Christ, if these
  things are not true; and if ye shall ask with a sincere heart, with
  real intent, having faith in Christ, he will manifest the truth of it
  unto you, by the power of the Holy Ghost.
  
  5 And by the power of the Holy Ghost ye may know the truth of all
  things.


With this in mind, the answer is that historical evidence can always be disputed.  There's an impressive corpus of "historical, archaeological, prophetical and other evidence and archaeological evidence" that testifies to the truth of the Book of Mormon, but like the evidence in favor of the Bible, it's all subject to interpretation.  It's all made up of things that happened to other people who are dead now. They can't tell their own story, and we never truly know if we have all the evidence or not.  New things are always turning up, and people come up with new theories and interpretations, sometimes based on reading the evidence, and sometimes based on reading something into the evidence that they want to be there.

But the witness of the Holy Ghost is testimony from God's spirit directly to yours.  It's a very personal affair, and it didn't happen to someone else.  It happened to you, and you know what you received from the Lord, and no alternate interpretation from some other person who hasn't experienced it can change the reality of that witness.  Simply put, it's the only truly trustworthy evidence there is, because it comes directly from the only perfectly trustworthy source there is.

As to how one is supposed to recognize the answer, this will probably sound like a cop-out, but the best answer is "you'll know it when it happens."  The problem is that any description requires a common frame of reference, and the witness of the Holy Ghost is a unique experience that isn't like anything that would be familiar to anyone who doesn't already have experience with it.  It's often described as a strong feeling of peace, and a sensation of warmth, a "burning in your bosom" to use the scriptural language, but it's not the same as physical heat.  But once one has experienced it, they truly know that they have received a testimony from the Lord.