Oh boy is this a question, and dont worry you arent the first to wonder. Hell even current and former soldiers have thought, how the Hell did this happen, especially when less than 2 years later he would be dismissed(intentionally on his part) from West Point for misconduct.
So lets lay down the basic facts, in late May 1827 Poe lies about his age and identity to enlist in Company H, 1st Artillery in Boston. He would be promoted to Sgt Major in January 1829, and was released 2 years into a 5 year enlistment that April in order to work towards acceptance to West Point which he would enter the following year, and then Spring 1831 would see him dismissed and his military career over for good.
In October 1827 the Company(along with the HQ of the 1st Aritllery) was sent from Boston to Ft Moultrie outside Charleston, and later Ft Monroe at the tip of the VA Peninsula guarding Hampton Roads. And Col House, CO of the regiment would be the one to promote Poe, and was personally familiar with him as they had been colocated for most of that time.
And despite his later personal demons, Poe actually made a fine soldier, was regarded as energetic, reliable, and bright. He also had something of a family legacy to live up to as a grandfather had been a quartermaster in the Continentals.
We can also note that about a year in Poe was promoted to Artificer, essentially the equivalent of a modern day Specialist. In that role he had responsibility for the care and condition of the artillery stores, and especially those new varieties of temperamental explosive shells!
So in the relatively flat enlisted org structure Poe was one of maybe 20 or so senior enlisted already with his company, which was itself only led by a 1st LT, and the CO of the regiment was present with him at the time.
It was the in late 1828 that Poe began to desire to leave the ranks, he revealed his deceptions to his Company Commander, a LT Howard. Howard then had Poe speak to Col House(and told Poe to write to his foster father), and shortly after Poe was reassigned to work with the regimental Adjutant as part of headquarters. Then just a few weeks later House exercises his authority in the matter and makes Poe his Regimental Sgt Major in January 1829.
Remember too that the Army was routinely short of men and had trouble filling the authorized establishment, or saw attrition to illness and desertion, with 75% authorized strength present not being a crazy baseline. An artillery company of the 1820's only had a max authorized force of 42 privates, 3 artificers, 4 corporals, and 3 sgts. The HQ then only had the Sgt Major and Quartermaster Sgt at least officially. So with such a small group, with the companies of the regiment rarely concentrated more than 2-3 to a post, it was easier for one to standout.
We honestly dont have hard explanations one way or the other. Poe was clearly a good soldier, and could now be honest about his slightly notable family history, was already a budding writer, and had some education from his stint at UVA. Those are traits that are certainly welcome in a senior enlisted in the peacetime artillery branch pre war. At least one source also mentions House was something of a lover of literature so there is another point of commonality for the two.
It was also about this time that Lt Col Worth rejoined the regiment from a stint at West Point and it could also be that his experiences there, and history of making the jump from enlisted to officer, are what pushed Poe over the edge to not just want to get out, but then go to the USMA. Their post at the time, Fortress Monroe may also have played a role. It was the closest thing to a home the artillery had, as it was home to the Artillery School. When possible artillery companies would be concentrated there for a year or so and trained up as a group, and new West Point grads would also spend a year or so there if they were destined for the artillery. So Poe would have had plenty of exposure to recent grads.
Worth seems to have been onboard with the idea as he helped advocate for his appointment as a cadet. And so with his relationship patched for the time being with John Allen, and his commanders in his corner, he was released from his contract that Spring.
Some easy reading on the matter! From the Army Space Journal. https://apps.dtic.mil/dtic/tr/fulltext/u2/a525759.pdf
Organizational history of the field artillery! Though Poe's old unit is actually now the 1st Air Defense Artillery Regiment, also having been designated the 1st Coast Artillery regiment, encapsulating the three pronged specialization the artillery branch eventually developed. https://history.army.mil/html/books/OH_of_FA/CMH_60-16-1.pdf
I'd like to add a bit of a broader perspective to the question, piggybacking a bit on u/DBHT14's answer. I think it's important to contextualize the size and scope of the United States Army in the 1820s, because it was, as an institution and as a part of American culture, much different than it is today.
Apart from the uniforms (which are somewhat outside the scope of the question), it's likely to be the size of the army that would be most shocking to a modern person. After the Treaty of Ghent, which ended the War of 1812, the congress of the United States restricted the size of the army to its pre-war levels to 10,000 officers and men. That was its authorized strength, the legal limit to the size of the army. In doing so, the army disbanded its dragoon regiments, collapsed its four rifle regiments into a single rifle regiment, but retained its corps of artillery. The infantry, at a wartime strength of fourty-four regiments, was brought down to eight, mostly by merging partially manned and wartime expansion regiments into newly reconstituted "fresh" regiments. The paper strength of the infantry was just over six thousand men total, with an additional rifle regiment with a paper strength of around 800 men. The small additional cavalry (a single regiment) and the corps of artillery would fill in the rest of the slots.
The problem was that the wartime strength of the regular army was around 30,000 by the end of the war, meaning that more than 20,000 men, mostly short-term members of the volunteer regiments raised for the war, would be expelled, along with 1,700 officers.
This may seem, living in the post-World War II event horizon, somewhat odd. But this happened after the War for Independence, as well, and it would happen again after the Mexican War and again after the Civil War (I talk a bit about this here ), and it should be understood as a reflection of the political ideas surrounding a standing army. To be short and blunt, standing armies - and soldiers, by extension - were viewed as pernicious outgrowths of tyrannical power. Or, conversely, they were viewed as manicured footpaths going directly from a well-maintained republic to a brutal tyranny. In either case, armies were viewed as moral wastes, as expensive and excessive embodiments of unrestrained power, as tools of political malefactors, and more or less turned every problem faced by a growing republic as one that needed force to solve. The lure and the supposed ease of using force was something that American politicians (and enlightenment thinkers in general) traced in the dramatic fall of the Roman Empire, and more recently, saw unfold in Revolutionary France. Massively reducing the size of the army after it had fought a war was an expected and a welcomed result of living in a society that was structured to maintain freedom for its citizens. Many modern writers tend to view this through a lens of disbelief or chagrin, but if we put our feet into the shoes of a citizen of the United States in the 1820s, this was neither surprising nor distressing, but a mark of a well-ordered society.
Those with military inclinations, like Poe himself, could find their place within local militia regiments. Poe, then 15, joined up with a local honor guard that formed and marched in the parade that welcomed Lafeyette to his home town in 1824. Local militias were meant to be reflections of the local reputation network, and tended to have "elites" - either men with money and means, or men with experience - elected as leaders, with the rest working to gain military (as well as social) skills for the betterment of their community, as opposed to the army, which was so easy to be misused for the benefit of a few alienated politicians. These were, at least, the ideals, but the model of the interested local militia and the aloof, mercenary regular army was breaking down even by the early 1820s. Again, more details and information here and here.
By January 1817, the army maintained a surprising portion of its manpower, with the secretary of war reporting that there were 10,024 officers and men in the United States Army, but even this was short lived, as in 1821, the size of the army was reduced, again, to a total strength of six thousand. Throughout the 1820s the army had difficulty maintaining even its paper strength. The artillery, which Poe joined, had a paper strength of 2,180 and in March, 1825, totalled 1,921. It was this army that Poe joined - a small, mistrusted force scattered across dozens of tiny forts and flung into border skirmishes (the Blackhawk War, the First Seminole War, etc) as they happened.
Life at the forts also would have had an effect on Poe's swift rise to Sergeant Major. Ranks and Posts were different things; artillery companies did not have First Sergeant positions, which were company appointments, but they did have post positions, like ordinance sergeant, which carried higher pay, but were limited in their authority to their job on the post. A sergeant major is a rough equivalent to first sergeant.
Poe's experience can further be qualified by the fact that, as a native-born American from a somewhat wealthy family, he stuck out like a bleeding wound in the enlisted ranks. Given the inherent suspicion of the American polity toward armies and soldiers, wealthy young men joining the army in peacetime were exceedingly rare. Most new recruits were immigrants, many of whom may not have spoken english, desertion was rampant, and discipline problems omnipresent. Which in a year of Poe's enlistment, the army not only experienced mutiny at West Point, but had a soldier murder another, and an attempt to murder an officer in a short lived Christmas Mutiny at Fort Mackinac, in Michigan. If he served consistently even for a short time, he'd very likely stick out to his officers, and justify his quick prominence within the tiny post life.
I could go on and on, but I hope this helped a bit. Happy to answer follow-ups!
All the numbers came from Gregory Urwins The United States Infantry, An Illustrated History.