How was Moses able to speak to over 600,000 people at once?

There are several ways. First of all, if God told Moses to speak to all the people, He would give him the ability to do so. At Pentecost, He gave the Apostles the ability to speak in tongues, and I would guess He gave Moses the ability to do something similar. Also, if Moses gave an important message, people would spread it.

The Greeks used early variants of megaphones made from bull's horns by 600 BC. It is possible that other cultures had similar tools even earlier.

My wife asked me a similar question. Having Moses speak God's commands/words to the Elders, and then to smaller groups; seems a quite reasonable explanation. We have a similar situation with Moses exhausting himself--settling disputes. Far too many people. His father-in-law gave wise counsel: Delegate! No need to do it all yourself. Regarding the supernatural explanation--I agree; this was a time of God making his immediate presence known to the whole house of Israel. (If one a-priori rejects the supernatural; one would reject the parting of the Red Sea; etcetera.) No problem with God using his own means to speak to these people in the hearing of one man. Although--that does seem like a case of "special pleading." We can only infer (conclude) that God chose a supernatural means of communication. (Directly to the people through Moses.) As an aside; the (thoroughly disproved) "Assured results of liberal 'Higher Criticism'" of the Old Testament claimed--without proof--that in Moses day; "people could not write." The spade work of biblical archaeologists put that unsubstantiated claim to death.

As Nelson Glueck states, on the one hand

It may be stated categorically that no archaeological discovery has ever contraverted a biblical reference”, whereas on the other “Scores of archaeological findings have been made which confirm in clear outline or exact detail historical statements in the Bible. [4]

Archaeologist William F. Albright observes:

The excessive scepticism shown toward the Bible by important historical schools of the eighteenth-and-nineteenth centuries, certain phases of which still appear periodically, has been progressively discredited. Discovery after discovery has established the accuracy of innumerable details, and has brought increased recognition to the value of the Bible as a source of history. [5]

There is much we don't know about the twelve tribes Exodus from Egypt. Trust God. The unbiblical claims of unregenerate men eventually fall flat. The same with science. Popular; sensationalized theories today may become tomorrow's relics. The Bible is like an Anvil. The hammers of godless men have been pounding away on the Anvil for thousands of years. The hammers wear out. The Bible remains.

[4] Nelson Glueck, Rivers in the Desert: A History of the Negev (New York: Farrar, Strauss & Cudahy, 1959), p.31.
[5] William F. Albright, The Archaeology of Palestine, pp.127-128, quoted by Josh McDowell, The New Evidence That Demands A Verdict (Nashvile: Thomas Nelson, 1999), p.61.

I would note that there is a set of skills and techniques, now largely lost because practice of them is largely unnecessary, by which (mainly) men in the past were able to speak to large numbers of people without artificial amplification. Abraham Lincoln and Edward Everett spoke at the dedication of the national cemetery at Gettysburg on 19 November, 1863, addressing about 15,000 people outdoors, without artificial amplification.

I don't mean to suggest that these means would make it possible for one person to speak to a single audience of 600,000, but I do expect that Moses might have been able to address an audience much larger than a modern reader might assume.