Still kind of lengthy but simplified:

Deep Battle Theory was something Soviet planners were thinking about since the 1920's, it was inspired by the Russian Civil War, in which combatants would often engage in fast paced warfare that sought to penetrate weak defenses quickly in order to gain a tactical advantage.

The idea was that warfare must move on from WW1 - for context, WW1 was fought across huge fronts and armies relied upon sweeping troop movements that always sought to deliver a decisive victory at the strategic level. If you think about it like a boxing match, WW1 was fought by boxers that always wanted to deliver haymakers to each other, huge crushing blows that they hoped would let them win by KO.

Soviets military theorists (and others) came to the conclusion after WW1 that this would no longer be possible given the size and scale of modern armies. You can't just KO an entire army in one movement like you could in the Napoleonic era, it never worked in WW1 and people certainly tried. So the idea was that you would use fast moving units (the Soviets initially thought cavalry would play a big role here, but we know how that turned out) to punch holes in the enemy line at various points and attempt to incircle them or cut off reserves/supply lines. The best example of this is when a line is broken in 2 or more areas and the attackers push far enough to then converge, creating a 'pocket' of enemy troops that have no supplies and no reinforcements. Do this enough and it starts to erode the enemy's capability to fight.

The whole 'theory into practice' issue arises because of Russian leadership purging the officer corps, meaning they were a little 'behind the times' on theories like this by the time Operation Barbarossa came around. This caused the Soviets to learn a lot on the fly and they did (after getting beaten bloody), often utilising concealment tactics (hiding supply dumps, not blanketing an area with artillery fire to 'soften it up' just before launching an attack, etc). This form of Soviet Deep Battle Theory, combined with the fact that German supply lines were stretched far too thin, made infamous encirclements like the one at Stalingrad possible.