Why did the Nazis use gas chambers instead of other methods of mass killing?

Well, gas chambers were arguably the most efficient method for mass extermination. This answer by the redoubtable /u/commiespaceinvader outlines how Zyklon B became the killing method of choice for the camps. Gas as a method of mass murder had already been used in the T4 program so there were experienced personnel available for the Reinhard camps. The SS also had experience with mobile gas vans which used redirected exhaust to murder Jews. There were debates within the SS over the efficacy of Zyklon B over carbon monoxide, but there was a general consensus within the SS that built up around late 1941 that gas was the best method available.

One of the salient facts of the Holocaust was that the gas chambers were the most efficient method available to the SS for mass killings. The Germans had experimented with various methods ranging from starvation, death through hard labor, and mass shootings. In the latter case, the Einsatzgruppens' operations in the USSR were destructive, but shooting operations could last for days. Moreover, mass shootings could not guarantee a wholesale elimination of the targeted Jewish population. These operations were time-consuming and used a large number of personnel. Centralizing and streamlining mass killing operations via death camps in Poland was a far more attractive option for solving the Jewish question. Not only would it allow the overcrowded Polish ghettos to be cleared out, these camps allowed for the deportation and extermination of Western Europe's Jews without creating undue stress on the German occupation. The camp apparatus within Poland involved only a small number of personnel given the numbers they killed. The camps also ran at a profit given Jewish property was now at the disposal of the Germans and their allies.

The real bottleneck for the death camps was not the gas chambers, but the disposal process for the corpses. The Nazis appreciated that the Allies could use the corpses of their victims as evidence if the Germans lost the war. Broaching the topic of defeat openly was a step too far for men like Himmler, but they did know that such evidence could be used against them. For one thing, the Third Reich had already done so with its discovery of the Katyn massacres of Polish troops by the Soviets. The Germans made a media spectacle out of the Katyn graves both to the wider European news agencies as well as for their own agitprop within Eastern Europe. Removing these mass graves would have prevented them being used to stoke revenge against a defeated Germany. This was in keeping with a longer practice pioneered in Germany's SS-run camps like Dachau where state-control of the bodies of those it murdered allowed it to keep the status of these individual indeterminate. Dachau and Buchenwald's own crematoria had demonstrated the utility of on-site crematoria as a means for the camps to control the narrative. Additionally, the initial mass burials also created a health issue as the corpses polluted the water tables and attracted vermin. The SS's Sonderaktion 1005 was tasked with finding the best methods of disposal of corpses. The engineering company J.A. Topf & Söhne patented a number of inventions for crematoria that would recirculate body fat and other heat so that corpses would assist in the cremation process.

The long and the short of it is the Germans certainly did experiment with a large number of methods to kill masses of people. Although the Germans did not abandon other methods of mass killing once the gas chambers were in operation, the use of poison gas in specially-constructed rooms very quickly demonstrated this was the most efficient method for the mass murder of Jews.