No one knows, and if someone claims they know, they are lying. We absolutely don't have enough information to know, although we can reconstruct with some confidence what contemporary people (those on the green or near it) thought happened, and why they acted as they did based on that understanding.
At around 4:30 on the morning of April 19, 1775, the sun is just starting to be visible in the town of Lexington, Massachusetts. Lexington today is a suburb of Boston, which trades heavily on its fame as the spot where the first armed conflict of the American Revolution took place, but in 1775 it's a country town, with a few buildings clustered around a town meetinghouse and a tavern. The town common is triangular, with a long side to the south; this is bounded by a road heading east towards Cambridge and west towards Concord. Another road splits off from the Cambridge road at the southeast corner of the common, heading more northerly and then bending north where it meets the road that runs southwest-northeast at the western end of the common.
On the southeast corner of the common are a belfry, in which a bell has been throughout the night tolling an alarm for the local militia to convene; the town hall, near but separated from the belfry; and a prominent tree that's the meeting point for the militia. Across the road to the northeast of the town hall is the Buckman Tavern, which in the manner of most colonial taverns has also served as a meeting place for "committees" of men, up to and including the provincial Committee of Safety meeting in defiance of the military governor's order. In the tavern, members of the militia have been periodically refreshing themselves throughout the night. On the western end of the common is a schoolhouse, and across the street from the west end of the common are a blacksmith shop and houses belonging to two men of the town.
About a third of a mile north of the common is the Clarke house, where revolutionary leaders John Hancock and Samuel Adams had until recently been staying -- warned that morning by messages brought by Paul Revere, among others, that the British regulars are out in force with their capture in mind, they have recently decamped for points further west.
As the sun starts to light up the Commons, a group of six light foot companies of the British army comes in sight of Lexington. The sight of the Regulars, though it had been predicted, startles the local militia somewhat -- though warned by Revere, they had sent out riders to see if the column was approaching, and when those men did not return, they were lulled into a false sense of security.
As the British infantry approaches, local militiamen under the command of Captain John Parker are forming a ragged double line on the western end of the Common, partially shielded from view of the infantry. If there were observant men at the front of the British column, they may have also spotted two men carrying a heavy wooden chest out of the tavern and through the line of militiamen -- John Lowell and Paul Revere had made a last-minute dash for a trunk that Hancock had left behind in the tavern, filled with correspondence pertaining to the Patriot cause. They took it west of the battle and disappeared into the woods nearby, intending to hide it there.
The local militia are forming, but there are relatively few of them on the ground -- Sylvanus Wood, a militiaman from Woburn, counted only 38 as he was falling into line, and most estimates say around 60 or 70 men stood their ground there that day. But from the British end of the field, the apparent numbers of the militia would have been increased by spectators milling around the Buckman tavern to the left of the militia formation and the horse stables and home to the south of the Concord road, to the right of the formation.
As Revere passed through the line, he heard Parker giving orders to his men to let the regular troops pass by, saying "Don’t molest them, without they being first." It was important to the Patriot leaders on this day, as it had been at Salem and other confrontations before, to not be seen as the first men to fire. Recollections of what was said were confused, but Parker chose to hold his ground there, even as some local men were arguing against it.
As the head of the British formation approached the Commons, they faced a fork in the road. The left turning would take them towards Concord, their eventual target that day, but would leave the militia on their right flank; the right turning would take them between the meeting house and the tavern, on the Buckman road, and head them directly toward the militia.
The Marine lieutenant leading the column, Jesse Adair, made a snap decision to turn right, and brought the three leading companies of Regulars along with him. Major John Pitcairn, in overall command of this detachment of troops, led his part of the line down the left fork and halted them there, then galloped across the Commons towards Adair's men.
The two leading companies of men with Adair, the light infantry companies of the 4th and 10th regiments of foot, follow him across the Commons and about halfway toward the militia. They deploy from column into line formation as they do so, one of the complicated military evolutions of the 18th century that British soldiers practiced to exhaustion. The British soldiers were cheering as they did so, adding noise to the already-confusing battlefield
Pitcairn, possibly with some other officers, rode up to the militia, pistol in hand, and ordered them to disperse. His words were recorded differently by several men there that day -- unsurprising, given the stress of the moment and the subsequent events. Some remembered him saying "Lay down your arms, you damned rebels!," while others remember something like "Throw down your arms, ye villains, ye rebels" or "Ye villains, ye rebels, disperse, damn you, disperse!"