What is the purpose of Macbeth's letter to his wife in Act 1, Scene 5?

In act 1, scene 5, Macbeth writes his wife a letter describing his interaction with the Three Witches, their seemingly favorable prophecies, and his new title as Thane of Cawdor. In the letter, Macbeth explains to his wife that the witches have supernatural power, and their prophecy about his new title as Thane of Cawdor was confirmed by Ross and Angus shortly after they disappeared into thin air. He goes on to write that the Three Witches also hailed him as future king after saluting him as Thane of Cawdor. The way in which Macbeth ends the letter indicates his purpose for writing it. Macbeth writes,

This have I thought good to deliver thee, my dearest partner of greatness, that thou might’st not lose the dues of rejoicing, by being ignorant of what greatness is promised thee. Lay it to thy heart, and farewell (Shakespeare, 1.5.7-10).

Macbeth reveals his love for his wife by referring to her as his "dearest partner of greatness" and says that he wants her to rejoice along with him in their promised greatness. Overall, Macbeth is excited, enthused, and hopeful about the witches' presumably favorable prophecy and wants his wife to rejoice along with him about their future positions as king and queen. He does not reveal his thoughts about plotting to overthrow the king or is seeking advice from his wife in the letter. Macbeth is simply sharing the good the news and wants the woman he loves to experience the same happiness and hopeful emotions that he is feeling at the moment.

In Shakespeare's Macbeth, there is a clear purpose to the letter our "hero" sends to his wife.

Macbeth, first and foremost, has wondrous news to tell Lady Macbeth. First, he met witches who offered him predictions. One has already come true, which means they will have more lands, money and honor in the King's court; the other promises that he will be King, and she will be Queen one day—it seems obvious he believes the old women after the first prediction came to be. In learning how excited he is about being Scotland's sovereign, we get the briefest glimpse of his "vaulting ambition"—his desire to become King that will eclipse everything else of value in his life.

We learn that Macbeth and his wife are very close. Not only because of how he addresses her, but also because he shares his news immediately rather than letting her wait until he returns home.

The third and perhaps most important purpose of the letter is to see how Lady Macbeth reacts to this news. We come to see how hard and calculating she is, and find that she worries that her husband is too kind to do what must be done if they are to secure the throne for themselves: which means murdering the King, for she sees no other way in which the crown can come to Macbeth.

Then she wishes Macbeth home quickly so that she can tell him what he needs to hear in order to do what he must:

Hie thee hither, that I may pour my spirits in thine ear. (I, v, 20-21)

It almost sounds as if she is casting her own spell: come quickly so I can pour my evil intentions ("spirits") into your ear, so you won't hesitate to follow through.

It could logically be argued that without Lady Macbeth's encouragement, Macbeth might have been satisfied with the rewards he had been given, and not kill the King: he says as much to his wife when he returns. In essence, this letter galvanizes the plot forward at the hands of Lady Macbeth's machinations.

Why is Macbeth's letter to Lady Macbeth in Act 1, Scene 5 significant?

The letter gives the reader an opportunity to get to know Lady Macbeth, her inner thoughts, what are ambitions are, what motivates her.  In this scene, we learn what Lady Macbeth thinks of her husband.

"It Is too full o' the milk of human kindness 
To catch the nearest way; thou wouldst be
great, Art not without ambition, but without 
The illness should attend it; what thou wouldst highly," (Shakespeare)

She tells us that he has ambition, but that she believes that he is too soft, too kind, not tough enough to go after what he wants.

She, on the other hand, is very tough, but is stuck in a woman's body and therefore must rely on her husband to take the lead and pursue her ambitious goal, to become king and queen of Scotland. She tells us that she will do her best to convince him to take the proper action needed at the opportune moment.

"That I may pour my spirits In thine ear, 
And chastise with the valour of my tongue 
All that impedes thee from the golden round, 
Which fate and metaphysical aid doth seem To have thee crown'd withal" (Shakespeare)

Lady Macbeth is so excited about the possibility of her husband becoming king that she goes outside to implore, to beseech the dark spirits to assist her, to fill her with power, she begs the universe to turn her from woman to man for this moment so that she can take the action needed, killing the king.

"Under my battlements. Come, you spirits 
That tend on mortal thoughts! unsex me here, 
And fill me from the crown to the toe top full 
Of direst cruelty; make thick my blood, 
Stop up the access and passage to remorse, 
That no compunctious visitings of nature 
Shake my fell purpose, nor keep peace between 
The effect and it! Come to my woman's breasts, 
And take my milk for gall, you murdering" (Shakespeare)

Lady Macbeth advises her husband on how to act in the presence of the king, so that he can manipulate the situation to his advantage, she tries to teach him how to be deceitful.

"Your face, my thane, is as a book where men 
May read strange matters. To beguile the time, 
Look like the time; bear welcome in your eye, 
Your hand, your tongue: look like the innocent
flower," (Shakespeare)