Section 131

I understood this a lot better when I learned that section 131 was actually received in three distinct parts at different times over two days (March 16-17, 1843).  These three revelations were later combined into one section by Elder Orson Pratt, acting under the direction of Brigham Young. 

Verses 1-4:
Joseph was traveling with William Clayton and staying at the home of the Johnson's.  Before they retired to bed, Joseph gave Brother and Sister Johnson some instructions on the priesthood.  Verses 1-4 were given in response to a conversation he had with William  Clayton that same evening.  We learn from these verses, that the celestial kingdom is divided into three degrees.  We learn that to qualify for the highest degree of the celestial kingdom one must enter into the new and everlasting covenant of marriage.  We are taught that those who comply with this law will have eternal increase.  What exactly is eternal increase?  According to the institute manual, "Those who comply with the new and everlasting covenant of marriage and endure faithfully to the end gain the right to become eternal parents."

Verses 5-6:
On the morning of March 17th, the Prophet Joseph discoursed on 2 Peter 1.  These two verses were given in that morning meeting.   Peter mentions the doctrine of having your calling and election made sure.  Joseph elaborates in verse 5.

Elder Bruce R. McConkie said:
“Those members of the Church who devote themselves wholly to righteousness, living by every word that proceedeth forth from the mouth of God, make their calling and election sure . That is, they receive the more sure word of prophecy, which means that the Lord seals their exaltation upon them while they are yet in this life. Peter summarized the course of righteousness which the saints must pursue to make their calling and election sure and then (referring to his experience on the Mount of Transfiguration with James and John) said that those three had received this more sure word of prophecy. ( 2 Pet. 1 .) . . .
“Those so favored of the Lord are sealed up against all manner of sin and blasphemy except the blasphemy against the Holy Ghost and the shedding of innocent blood. That is, their exaltation is assured; their calling and election is made sure, because they have obeyed the fulness of God’s laws and have overcome the world.” ( Mormon Doctrine, pp. 109–10; see also D&C 132:17, 26 ; Notes and Commentary for D&C 76:53 .)

Verse 6 talks about gaining knowledge.  The manual points out that although this verse is true of all knowledge, it is important to remember that this revelation was given in connection with verse 5.  No man can be saved without having this sure knowledge--the sure word of prophecy.

Verses 7-8:
Joseph attended a lecture given by a Methodist preacher on the evening of May 17th.  These two verses were received afterwards to offer some correction to what had been said.  I don't have a written account of the preacher's lecture.  But apparently he somehow addressed the concept of matter.  Scientists in Joseph's day believed that matter could be created and destroyed.  In more recent times, scientists have proven that matter can not be created and destroyed.  Matter and energy are just different forms of the same thing. Matter is eternal.   
Orson Pratt once said:

The materials out of which this earth was formed are just as eternal as the materials of the glorious personage of the Lord himself.