Deceptive spring of 1943
The general situation on the Eastern front in late February, early March of 1943 was this: the German summer offensive Fall Blau have failed, and the Germans did not capture desperately needed oil. The Soviets have counter-attacked with Operation Uranus and later with Operation Little Saturn, destroying the German Sixth Army in the process and parts of the Fourth Panzer Army, plus a large part of the Romanian, Hungarian and Italian troops on Soviet soil. Italians and Hungarians withdrew their remaining forces from the USSR, and the Romanians were confined to Crimea and Kuban bridgehead.
Practically, only the remaining Axis power (besides the Germans) with a significant army in USSR was Finland, in the far north. However, the Soviets have suffered significant casualties both in 1941 and 1942, especially around Rzhev. The Germans counterattacked in the Third Battle of Kharkov, and managed to retain parts of the territory captured during the summer (the Crimea as the most significant gain). Now, the usual spring rasputitsa dictated a pause in major operations.
German reasons for peace talks: For the Germans, it became clear that the USSR could not be knocked out in one blow (Operation Barbarossa) or destroyed economically (Fall Blau). Furthermore, with their Axis allies withdrawing from the fight, long-range operations with stretched frontlines were out of the question due to a lack of manpower to man those long lines.
At best, offensive operations with limited goals like Zitadelle could be launched to destroy Soviet forces, not to capture vast territory. The perspective of war in the East was long war of attrition, with German perceived superiority in tactics and technology being balanced by Soviet numerical superiority and ruthlessness. On top of that, US forces have landed in North Africa, and their bombers started raiding German occupied Europe, so from that side increased pressure could be expected in the coming months.
Soviet reasons for peace talks: Soviets had their own reasons to possible conclude a separate peace with Germany. First, large parts of USSR remained occupied, therefore limiting both conscription potential of the Red Army and the potential of the Soviet war economy. Especially dire was the food situation, as most fertile lands remained in German hands (Ukraine).
Second, the Red Army was still not German's equal in tactics and operational art, which the recent defeat at Kharkov clearly showed. It would take months and years to finally defeat Germany, and victory will be paid in blood of millions. Third, Soviets (especially Stalin) doubted good intentions of Western Allies. British and Americans were reluctant to open a significant second front - during the Casablanca Conference they declared willingness to fight until German unconditional surrender, but besides that, there were little concrete measures taken, except continuing the fight in North Africa and even that appeared to be stalling (defeat at Kasserine Pass).
For the Soviets, and especially Stalin, all that seemed like pinprick in the war effort, and from their spies in British ranks they knew that Churchill preferred to have the Germans and Soviets at each other throats in order to block a communist expansion into Europe. Finally, in the period mentioned, even convoys with Western Lend-Lease for the USSR were periodically suspended (August 1st - August 31st, 1942 and March 1st - July 31st, 1943), giving another reason for suspicion.
Japanese angle: There was another party greatly interested in peace between Germany and USSR, and this was Japan. Japanese interest in all of this is simple to understand - Japan was not in the war against USSR, but it was in the war with the US and UK. By establishing peace between Germany and USSR, the Germans could now devote more resources to fight against the Western Allies, thus helping the Japanese situation which was deteriorating after the loss at Midway in 1942, and the costly Guadalcanal campaign. According to Russian documents, Japan started their mediating attempts in early 1943, but only in September of 1943 did the Japanese ambassador formally offer their service to Molotov, which was rejected by the Soviets - by September, the Soviet offensives were already rolling the German army towards the Dnieper.
Italian angle: The Italians also had their own reasons for wanting a separate peace between Germany and USSR. The Italians had no particular interest in the East (their primary concern was the Mediterranean and North Africa). With the loss at El Alamein and Torch landings, the Italians saw no reason to continue fighting in the East, especially since their own forces suffered catastrophic losses. According to some Russian sources, Mussolini himself pressured the Germans to conclude a separate peace with Soviets. Again, the name of Peter Kleist is mentioned, along with contacts with Soviet diplomats in neutral Sweden.
Talks in Sweden: What many sources consider as possible (unlike story of Molotov and Ribbentrop meeting somewhere in occupied USSR) are contacts between the Soviet ambassador in Stockholm, Alexandra Kollontai and her peer on the German side Hans Thomsen in March and April of 1943. This would be somewhat usual practice; envoys from two warring sides meet in neutral country for discrete talks.
They would present initial positions of their respective leadership and throw around some informal proposals. Of course, if there is some progress, things could escalate to the level of foreign affair ministers. From what we know now, there was no such success in these talks, and they were broken in May. Some sources mention renewed Soviet interest in June, just before Zitadelle, but that was it.
Possible reasons for failure: Since the talks themselves are not confirmed, possible reasons for the failure remain speculative. From the Soviet side, supposedly Stalin only toyed with the idea in order to get more concessions from the Western Allies. He even allowed their intelligence services to learn about talks in order to "frighten" Churchill and Roosevelt. On the German side, again supposedly, Hitler did not want to negotiate without "big win", i.e., from the position of weakness after Stalingrad. That big win was supposed to be Zitadelle, but since that did not happen, negotiations failed.
In any case, the window of opportunity was small (a few muddy spring months) and when this did not happen, Soviets had fewer and fewer reasons to negotiate as their armies were now decisively advancing across the Eastern front. Since from August 1943 the initiative in the war passed firmly to the Soviet side, and the Allied help was getting more substantial, the only remaining option was the complete defeat of the Third Reich.