According to the LDS church, why was Jesus Christ born when he was born?

I've read through the linked definition of "Meridian of Time" several times, and I think the answer is quite simple.

The question is looking at it from the wrong angle. It's like asking "why are all the sided of a square the same length?" or "why does a triangle have three sides.

Looking at this portion of the article you linked to:

The meridian of time may also be seen as the high point of mortal time. Latter-day revelation shows that all of the ancient prophets looked forward to the Messiah's coming (Jacob 4:4; Mosiah 13:33-35;15:11). His coming fulfilled their prophecies, and he was prefigured in the Law of Moses (Mosiah 13:29-32) and in ancient ceremonial ordinances (Moses 5:5-8). The meridian of time is the apex of all dispensations because of the birth, ministry, and Atonement of Christ. Without him all prophetic writings and utterances would have had no efficacy, and the hopes of mankind today and forever would be but futile desires and yearnings without possibility of fulfillment. MARSHALL T. BURTON

The "Meridian of Time" exists only because Jesus came at that moment.

His coming at that time is what makes it the"Meridian of time".

To my knowledge there is not a clear answer to this question in standard LDS church doctrine. It is likely that many church leaders over the years have given their opinion on the matter but whether these statements constitute doctrine is questionable.

Mormons do believe is a purpose for Christ being born at this time, as they believe there is a purpose in everything God does. It is possible that the setting in Jerusalem described by the LDS apostle Elder James E. Talmage was the most ideal for the life of Christ. For example, it would be much more difficult to perform miracles without attracting dramatic international attention in the modern age. If Jesus came thousands of years earlier, how many records of him and his teachings would have survived? Assuming the society that puts him to death so he can die for everyone's sins acts of their own free will, how many societies are there that would execute someone for claiming to be the son of God? These are some common sense lines of thought that might explain the timing.

If you're interested in more specifics, Jesus the Christ is perhaps the most detailed Mormon treatise on the subject.

Not being familiar with the term "meridian of time" I clicked on your hyperlink and read the definition given there. While there are many terms Christians use today which are not in the Bible, including "the meridian of time," I can only suggest there is a term in the Bible which tells us why Jesus came when He did. It comes from Galatians:

ὅτε δὲ ἦλθεν τὸ πλήρωμα [pleroma] τοῦ χρόνου [chronos], ἐξαπέστειλεν ὁ θεὸς τὸν υἱὸν αὐτοῦ, γενόμενον ἐκ γυναικός, γενόμενον ὑπὸ νόμον (4:4)

In other words, "when the time had fully come," or "at the appropriate time," or "in the fullness of time," that was the time God sent His Son,

ἵνα τοὺς ὑπὸ νόμον ἐξαγοράσῃ, ἵνα τὴν υἱοθεσίαν ἀπολάβωμεν [born of a woman, born under the Law that He might rescue those under the Law, that we might receive the full rights of sonship]

So we have the "fullness of time" and the "full rights" of sons, both expressions fitting in nicely with the theme of Galatians, which is the fullness offered by Christ, versus the emptiness offered by the Judaizers. Christ offers freedom; the Judaizers offer slavery. But I digress.

The importance of prophecy, as you indicated in your question, must not be underestimated. While God may inhabit eternity, He reveals Himself in time and space. That was His plan from all eternity. That plan comprises part of the eternal decrees of God. God is the only One who works out all things according to the purpose of His will (Ephesians 1:11).

God, as the Lord of time, knows the end from the beginning and the beginning from the end. As both omniscient and omnipotent, God not only can do (i.e., He has the power to do) what He has decreed He will do, but He also will do what He has decreed, and in the best possible way (i.e., He has the wisdom and knowledge to "pull it off").

There were other, more prosaic--albeit important--factors at work when Jesus appeared on the scene at the appropriate (or the fullness of) time:

• The Pax Romana, or Roman peace, which was characterized by an epoch of relative calm made possible by the sheer power, not to mention the administrative expertise, of the dominant world power of the day, imperial Rome.

• The ubiquitous presence of the Greek language, attributable to the tremendous admiration the Romans had for all things Greek. The dissemination of the message of the Christian Gospel was aided in no small measure by the existence of Greek as a significant medium of communication.

• The improvement of transportation throughout the Roman Empire, thanks to the wealth of Rome, not to mention, again, its administrative ability, its military might, and ready access to slave labor to do the dirty work in creating roads, some of which are still in existence today (e.g., the Applian Way, an ancient Roman highway extending from Rome to Brundisium--now Brindisi, which was constructed beginning in 312 BC by Appius Claudius Caecus, and was about 350 miles long). The better the infrastructure of transportation, the more readily the good news could spread to the far corners of the empire.

• The spiritual state of Judaism at the time Jesus burst on the scene. If ever there was a time for a nationwide revival within the geographical heart of Judaism in Israel, it was the first century of the common era. That is why the forerunner of Jesus, His cousin (John the Baptizer), preached a message of repentance to prepare the way of the Lord. John bore witness to that light which was coming into the world as a testimony not only to the Jews but also to the Gentiles (see Luke 2:32, 38). We must remember that God's covenant with Abraham was a far-reaching covenant which would touch with blessing all the nations of the earth, starting of course with Israel.

• The receptivity of the people to the message of Jesus, due in part to the bifurcation within the Roman Empire between the "haves" and "have nots". If ever there was a time for a Redeemer who would bring a message of hope to the underdogs of society, it was in the first century (see Luke 4:14-21 in this regard).

• The need for God to fulfill not only His prophecies, but also His promises.

All of which brings us back to the "fullness of time." God is a God of His word. His word is His bond. What He says He will do, He will do, and it will be done in His time, according to His timing, and in the best possible way. In short, the "who, what, when, where, how, and why" of Jesus' birth were perfectly orchestrated by God well in advance, right down to the minutest detail. (Even Martha Stewart would be proud of how perfectly things were laid out, if only her heart were directed toward what really matters in life!)

At this time of year, when we celebrate the birth of baby Jesus--Emanuel, our Savior and our Lord, it is comforting to know that God's purposes and decrees could not be thwarted, neither by a Jewish king called Herod, whose heart "the prince of the power of the air" filled with hatred for the true King of the Jews, nor by an emperor, Caesar Augustus, who decreed that the then-known world be taxed, which only served to bring the parents of Jesus to the City of David, Bethlehem, the "house of bread," where a child would be born who would one day declare Himself to be the bread of life (John 6:48 ff.).