The deep battle, and later on Deep operation doctrine was developed as a concept in the inter-war period, with the most prominent proposer and theorist being Mikhail Tukhachevsky. After the Red Revolution when the soviets took power, the infant soviet state faced many threats. Foreign forces and the new White Russians threatened to destroy the new government and the new red army was ill-prepared to face such threats with its resources exhausted after 3 years of war.

The result was the Russian Civil War (1918-1921), a formative experience for both the Soviet State and its Red Army. During 1918 and 1919, Lenin and his commissar for military affairs, Leon Trotsky, used the railroad lines to shuttle their limited reserves from place to place, staving off defeat time after time. This became known as echelon war, in which large forces where shifted on internal lines by railroad train (echelon) to reinforce successively threatened fronts. Some infantry divisions shifted between fronts as many as five times in the course of the war. This experience gave all participants an abiding sense of the need for strategic reserves and forces arrayed in great depth.

Because of this experience, Soviet military commanders developed a unique view of warfare. In the West, the trench stalemate of WW1 dominated most military experience, albeit reformers sought solutions to that stalemate. The eastern front, being longer and less well fortified, had never been as rigid as the trenches of France. More importantly, the Russian Civil War was characterized by vast distances defended by relatively small armies. Soviet commanders tried to integrate all tactical operations into an overall campaign plan, aiming for objectives deep in the enemy's rear. The two keys to victory proved to be concentration of superior forces to overwhelm the enemy at a particular point, followed by rapid maneuvers such as flank tactics, penetrations and encirclements to destroy the thinly spread enemy. Such maneuvers required a highly mobile offensive force, which in the civil war consisted of armored railroad trains and cars together with horse cavalry.

During the interwar period, the brilliant Civil war commander Mikhail N. Tukchachevsky and military theorist Vladimir K. Triandafillov developed a strategic theory of succesive operations based on the Soviet military failure against Poland in 1920 and the failed German offensives in France during 1918. Put simply, the believed that modern armies were too large and resilient to be defeated in one cataclysmic battle. Instead, the attacker would have to fight a series of offensive battles, each followed by a rapid exploitation into the enemy rear and then another battle when the defender reorganized his forces. To place these battle in a common strategic context, Soviet soldiers began to think of a new level of warfare,connecting the tactics of individual battles with the strategy of the entire war. This intermediate level became known as Operational Art (operativnaia iskusstva).

Operational Art is the realm of senior commanders who plan and coordinate operations of large formations within the context of an entire campaign; that is, a series of actions culminating in the achievement of a strategic objective. In 1927, Svechin summarised it as such : "Tactics makes the steps from which operational leaps are assembled, strategy points out the path".

During the 1920's and 1930's ,Soviet theorists perfected the tactical concept of Deep Battle (glubokii boi). They planned to use new technology, especially tanks and aircraft, to penetrate the elaborate defense systems of the world war.

By 1936, technological change led, in turn, to the larger concept of Deep Operation ( glubokaia operatsiia). Instead of planning to penetrate the enemy in a single deep battle, theorists projected penetrations and exploitations to an operational depth of 100 kilometers or more. The essence of such a deep operation was to use modern weapons at one blow to neutralize all of the enemy's defensed to the maximum possible depth and then exploit so rapidly that the defender could not reorganize his units. This meant simultaneous operations at various points, all carefully coordinated. As Egorov wrote : "The principal and basic task of military art is to prevent the formation of a firm front [by the defender], imparting a destructive striking force and a rapid tempo of operations."

Initially, this would be achieved with the weapons of the Russian Civil War : infantry, artillery, and cavalry supplemented by armored cars. Yet, Soviet Theory evolved rapidly, and by the early 1930's, Red theorists included the entire spectrum of mechanized forces functioning as a sophisticated combined-arms team. Infantry, led by tanks and supported by artillery and engineers, would penetrate the enemy's defenses while other artillery and aircraft struck deeper into the enemy's rear, to be followed by large, independent airborne and armored formations. Tanks would therefore be organized into three echelons: some tanks would lead the infantry penetration, others would conduct short-range exploitation of the resulting breakthrough, and still others, operating in large combined-arms mechanized formations, would pursue and encircle the beaten enemy. These concepts were codified in the Red Army's Provisional Field Service Regulations of 1936 (PU-36).

The idea of a deep, mechanized operation was not unique for its time. Military theories of all major armies evolved in the same general direction, using varying degrees of mechanization to penetrate enemy defenses and thereby avoid the stalemate of trench warfare. What was unprecedented about the Soviet concept, was the official sanction it received from the Soviet dictator. Stalin dedicated a large portion of his first five year plan to providing the industrial capacity and production needed to implement that concept.

In the mid 1930's the Soviet Union lead the world in the production, planning and fielding of mechanized forces. It was well ahead of its German counterparts in both theoretical concepts and practical experience of armored warfare. In Germany, Heinz Guderian and other armored theorists received only limited support from civilian and military leaders. Even by 1940, large operational employment of panzer forces was still an experiment.

Unfortunately, this advantage evaporated in 1939, largely due to Stalin's officer purge. An entire generation of commanders, government administrators and factory managers was decimated. Younger men, often lacking combat experience or training for their positions, found themselves thrust into high command.

In July 1939, a special commission reviewed the entire question of armored force organization and few experienced armored officers were allowed to participate. The commission reached a compromise that directed the removal of motorized infantry elements from tank corps and brigades, reducing such units to an infantry support role.

In 1941, due to the military disasters incurred during Operation Barbarossa, the Red Army had temporarily simplified its force structure, and it was unable to implement its prewar military theories because it lacked the initiative, weaponry and the trained leaders to employ that structure and those theories effectively. After Stalingrad and Uranus, however, the Soviets had accumulated enough of these scarce commodities to begin elaborating on their force structure and updating their doctrinal concepts. Tested in late 1942 and 1943 and perfected in 1944 and 1945, the new Red Army developed rapidly to the point where it could conduct the ideal Deep Operation.

In 1943, the Red Army needed a larger mechanized force formation, analagous to a panzer corps or panzer army, for deeper, operational exploitations up to 500 kilometers. The result was the 1943 tank army. Unlike the 1942 organizations of the same name, which where a mixture of tank, cavalry and rifle units (soviet name for infantry units), the new tank armies were designed to share a common level of mobility and armored protection.

For the remainder of the war, the five (later six) tank armies were the spearhead of Soviet deep attacks, conducting operational maneuver and seeking objectives deep in the German rear areas. On a map, Soviet offensive plans often resembled a set of Russian nesting dolls, with shallow encirclements inside of other, deeper encirclements. The separate tank and mechanized corps, sometimes replaced by cavalry-mechanized groups in difficult terrain, were attached to the forward combined-arms armies (former rifle armies, the bulk of the army) so that they could encircle one or more German corps immediately behind the German main defense line. Meanwhile, operating under front control, the tank armies bypassed these struggles, straining to penetrate as far as possible into the operational depths and thereby achieve larger encirclements.

Such operations were first seen at Stalingrad in 1942 and were the main Soviet doctrine employed during the 1943-1945.

Source: When Titans Clashed by David M. Glantz & Jonathan M. House

Edit : fixed some typos

This answer from a couple of years ago has a pretty good explanation, unfortunately the commenter has deleted their account but the great /u/Georgy_K_Zhukov posts a comment in there with book recommendations!