Well, this is complex, but really there are multiple reasons, primarily that the military high command wasn't entirely purged and most of who were weren't actually that attractive to the common soldier, but let's analyze the issue.

First who were the heads of the military in 1937 and who of them were purged? The first posting that needs to be mentioned is the People's Commissar of Defense, who technically acted as head of the army. In 1937, this was a lesser known man named Kliment Voroshilov, who had known Stalin since 1906, and who was in a deep friendship with Stalin, allegedly even surviving screaming at him in 1939, over the poor performance of the Red Army in Finland, which Stalin had blamed on him and for which he blamed the purges. Voroshilov was inept, vain, and entirely incompetent for leadership in warfare more complex than the archaic tactics of the Russian Civil War (which had seen such habits as a complete lack of radio producing heroic field commanders who led their soldiers into charges against the enemy, Voroshilov would repeat this outdated habit during the defense of Leningrad). More importantly however, he was politically reliable. He was trusted, and unlike other commanders, the simple, likely poorly educated or completely uneducated soldiers comprising the base of the Red Army felt more connection to this East-Ukrainian son of peasants then they did to tsarist educated commanders, who were the bulk of the first wave of purges.

Now, the most notable commander to be purged, in the first wave of arrests that affected the military no less, was Mikhail Tukhachevsky, deputy of the People's Commissar of Defense. Though well trained and an expert theorist who has been claimed to have made the Red Army the most advanced military of the time, Tukhachevsky lacked the favor of both the bulk of the army and the political elite which by then had come under Stalin's exclusive control. Tukhachevsky was also known to have had a sort of rivalry with Stalin, stemming from the polish soviet war in which both were high ranking commanders. The 'competence divide' in the Red Army was notable, with some lesser trained officers reported to have complained to Voroshilov when Tukhachevsky and his clique mocked their lack of education.

Tukhachevsky was arrested mid-1937, and alongside 7 colleagues who shared his fate, he was tried in secret, sentenced for conspiracy against the soviet state, and executed just as secretively. This was significant, because the politically unreliable members of the highest authority were immediately removed. So, in this way, those who were the most likely to organize a fight were already gone, even if they hadn't been as popular as stalinist commanders in the first place.

Now, of course, there's a multitude of other reasons why the military put up no fight, and I'll try to go by importance when listing these.

Distrust permeated every layer of the Red Army command, including the highest stratum. Throughout the bulk of the military purges and the 'great terror' phase, Voroshilov was the supreme authority in matters of the army, and he had been notoriously happy to contribute to the purges. This manifested most prominently in his eagerness to spread rumors prior to the arrest of his deputy (he had despised Tukhachevsky), but nearly all arrest lists concerning the army bore his signature. One thing to note here is that Voroshilov did put significant effort into saving many petty commanders at the time, and he did mitigate the impact of the purges. Now, back to the topic at hand. After Tukhachevsky had been cut off, the most dangerous element, the person who would have been most likely to fight, had been removed quickly and efficiently. All that remained is a system of higher commanders that were either loyal to Stalin or could trust nobody to share their disloyalty.

This, of course, was even more remarkable on lower levels. Not a single historical account comes to mind of a society that was as thoroughly permeated by the activity of the secret police as the USSR under Stalin. The NKVD was omnipresent in all aspects of life, meaning also that the military had significant rates of collaboration, usually intent on stealing away authority posts, personal revenge, or the occasional genuine political concern. Thus, lesser officers, the main victims of the purge, were internally unreliable, cross-reporting amidst themselves and offering no viable way in which resistance could be organized, which stood as a testament to the efficiency of the characteristic NKVD habit of arresting those who had yesterday accused their colleagues of conspiracy.

So, though more can always be said, I'll sum up what I believe to be the most important factors in creating a lack of military resistance in the Red Army. First, major positions of power were occupied by stalinists who were happy to enable at least some major sections of the purge. The first, largely unexpected wave of arrests hit the most dangerous persons, removing the most important risk factors. Beyond them, the army was generally prone to internal quarreling which resulted in no significant anti-state organization, ironically insofar as conspiracy was the first and most important accusation used by the soviets to purge their military.