This is an odd question to answer, because the Red Army wasn't particularly successful during 1941 and 1942. Their troops were inexperienced, they were in the process of modernising, some of their best leaders had been purged and the doctrine they had written had been discredited. However, by 1943-44, these problems had been solved. Both the troops and their leaders had gained experience, their equipment was some of the most modern in the world, and their strategic ability was second to none. The contribution from lend-lease was significant, but was mostly focussed on enabling Soviet logistics.

In Deep Battle, the Soviets had a highly sophisticated doctrine. In the offence, it called for an initial broad-front attack, to create a breakthrough. A second echelon would then exploit this breakthrough on a tactical level. A third echelon was present to exploit the breakthrough on an operational level, or to reinforce the second echelon as necessary. In defence, the first line would delay and attrit the attacking force. The second echelon would counter-attack the attacking spearhead, while the third remained able to launch a counter-offensive, or reinforce the second echelon as necessary. It was a highly difficult doctrine to counter, requiring a combination of a thick, wide defence with counter-attacking forces. As Kursk showed, attacking a force following this doctrine was nearly impossible.

Deep Battle wouldn't have been as successful as it was without the Soviet strategic ability. They were able to concentrate force effectively, conceal the fact that they had concentrated that force, and supply it as it moved. The Soviet attack on Manchuria in 1945 is an excellent example of this. They were able to move nearly a million men along the Trans-Siberian railway, while concealing this fact so successfully from the Japanese that they thought an attack would be impossible until 1946. The troops in Mongolia were supplied effectively despite being hundreds of kilometres from their railheads. The Red Army had units called Machine-Gun Artillery units. These units were set up specifically to hold areas of the front that had been stripped of front-line troops to bolster an attack elsewhere. Soviet deception operations, called maskirovka, were sophisticated,and took place across the front. During the run up to Operation Uranus, troops around Stalingrad constructed fake defensive fortifications. Pontoon bridges across the Don were constructed in the Voronezh Front's sector, to draw attention from those being constructed nearer to Stalingrad. Troop movements near Moscow were made obvious to the Germans to draw their attention away from Stalingrad. Soviet logistics benefited most from lend-lease. Most of the trucks and trains that shipped supplies forwards had been supplied by the US, as had much of the supplies they carried.

Finally, the Red Army was very well equipped, particularly in artillery and armour. Each rifle division was supported by 52 field guns, 136 mortars, 66 AT guns and 16 SPGs. This was commonly added to by attached units. German units had only 48 artillery pieces as organic support. Given the Soviet strategic abilities detailed above, it was not uncommon for Soviet attacks to be supported by up to 420 artillery pieces per mile of front. The Soviets operated several artillery corps for supporting such attacks, with 800 artillery pieces and a similar number of Katyusha multiple rocket launchers. Soviet tanks were some of the most capable tanks of the war. The T-34-85 was fast, well armed, and decently armoured. It was, if well handled, capable of going toe-to-toe with Panthers, and was generally superior to the Panzer IV. The IS series of heavy tanks was heavily armoured, with a 122mm gun designed for infantry support. The typical Soviet armoured unit, the tank corps, was equipped with 228 tanks and 42 assault guns. By comparison, German units had at most 200 tanks and 45 assault guns. Given German supply difficulties, it was rare to encounter a German armoured unit at full strength.

For more information on the Red Army, I'd recommend the works of David Glantz. His work August Storm: The Soviet 1945 Strategic Offensive in Manchuria, found online here, contains a well-executed analysis of the Red Army's strengths.

/u/thefourthmaninaboat does an excellent job of talking about the material and strategic aspects of the Red Army, but I also think it is worth noting that the psychological shift for the Red Army in the Winter of 1943 should not be undersold. The confidence and morale of an Army is an important underlying factor in the ability to maintain offensive operations, as well as hold defensive lines. While the material and strategic improvements helped buoy this, the propaganda campaign that Soviet citizens, and Red Army soldiers in particular, experienced from the Kremlin was also important in shifting their attitudes. The early battles of 1941 compared to the fighting spirit of the 1943 army are night and day, and largely thanks to Stalin convincing the people that they were defending their country against evil invaders rather than defending an uncaring regime. The German treatment of prisoners of war, and the captured territory certainly helped build an almost fanatical hatred of the Germans.

While this ultimately helped contribute to better fighting spirit, the negative consequences in East Prussia and Berlin show how this type of fervor, once built, is hard to contain.