On the topic of marriage in heaven, how does the LDS church reconcile Matthew 22:23-33?

It's important to understand who 'they' refer to. From verse 23 we know the Lord is talking directly to the Sadducees, which say that there is no resurrection. They were asking a question to bait/trap the Lord in contradicting Moses, the lawgiver (see Deut 25:5-10). Remember this question is about what happens in the resurrection when the Sadducees don't even believe in the resurrection (see vs 23). The Savior effectively dismissed their question on marriage by stating that in the resurrection they neither marry, nor are given in marriage, but are as the angels of God in heaven(vs 30). Jesus is trying to turn to the important issue of the Sadducee's unbelief in the resurrection (see vs 31-32).

So here are some important points on the Lord's statement:

1. First, we see that it was made in response to an attempt by the Sadducees to trap the Lord. Consequently, it would not have been the Lord’s final word on the subject. Why should the Lord scatter pearls before them that they would only trample underfoot?
2. Second, the Lord did not say there would be no people in the married state in the resurrection, but that there would be no marriages made in the resurrection.
3. Third, we must be clear about the “they” who are neither marrying nor being given in marriage. The context of the scriptures just cited suggests a generic rather than a specific meaning. Simply put, that means no marriages are made in the resurrection. The Lord was warning the Sadducees. They were Jews of the day who had rejected him and therefore had no access to the higher ordinances of the Melchizedek Priesthood. How could these men, whom Jesus had called a “generation of vipers” (Matt. 3:7), qualify for the highest blessings of the celestial kingdom?
4. The Lord has previously, and we believe has again in this day, given the power to bind on earth and in heaven (Matt. 16:19, Doc&Cov 132:19)

Joseph Smith has said:

“A man must enter into an everlasting covenant with his wife in this world, or he will have no claim on her in the next. “No man can obtain an eternal blessing unless the contract or covenant be made in view of eternity. All contracts in view of this life only terminate with this life. [Such is the] case of the woman and seven husbands. Those who keep no eternal law in this life or make no eternal contract are single and alone in the eternal world and are only made angels to minister to those who shall be heirs of salvation, never becoming sons of God, having never kept the law of God, that is, eternal law.” (emphasis added, in this answer referencing Sadducees who did not live the new law that Jesus taught)

Similarly, Joseph Fielding Smith said:

“So here we have the true understanding of the Savior’s answer to the Sadducees. If by any chance, any who believed as they believed, and therefore were married for time only, proved worthy to obtain that world, that is the kingdom of God, they would have to enter there separate and singly to become servants-angels-to wait on those who were worthy of the exaltation. These Sadducees who might be worthy of a place in that kingdom would be in exactly the same condition that members of the Church of Jesus Christ will find themselves, if they likewise, have been content with a civil union only.” (emphasis added)

On marrying, James E Talmage sums up the answer by saying:

“In the resurrection there will be no marrying nor giving in marriage; for all questions of marital status must be settled before that time, under the authority of the Holy Priesthood, which holds the power to seal in marriage for both time and eternity.”

References/See also

• Ensign 1986- 2nd article (heavily referenced)
• Matt 19:4-7- Jesus taught Pharisee, who believe in resurrection, about marriage
• Doesn't Matthew contradict Eternal Marriage- answer to a similar question
• Joseph Smith's Commentary on the Bible, pg 125
• Answers to Gospel Questions Vol. 5, pg 118
• Jesus the Christ chapter 31

--EDIT--

Specifically, on the topic of eternal marriage, the LDS believe that when a man and woman are married in the temple, their family can be together forever. The article you mention says:

Mormons assert that for a celestial marriage to occur, several key steps must be followed. First, the man and woman must be sealed, or bound, by one holding the authority of God to perform such sealings, which can only take place in Mormon temples. Second, the couple must seek, with all their hearts, to individually and jointly follow Jesus Christ. Third, the sealing must be confirmed by God through His Spirit.

Biblical basis of sealing/binding power is seen in Matt 16:19 where Jesus Christ gives Simon the power to bind on earth and in heaven. The LDS believe that that authority was lost during the apostasy and has been restored through the prophet Joseph Smith. (see also Eccl 3:14, Mark 10:9, Matt 28:18)

Eternal marriage is another doctrine that central to God's plan. Matt 19:4-7 says, What therefore God hath joined together, let not man put asunder. When a man and woman are sealed together in the temple the LDS believe they are sealed eternally, that their marriage isn't til death but will last forever. (See also 1 Cor 11:11, Ephesians 5:31)