Bavaria was and, at the same time, was not unique in the post World War I era. 

Like in many other German states, people lost their faith and trust in both the political system and the monarch. Ironically, the Bavarian king supported the idea of a Siegfrieden (peace through victory or victorious peace) along with the annexation of new territories for Bavaria. He behaved in a militaristic way and was perceived to be too close and friendly with Prussia, i.e. Wilhelm II. Bavarians rejected the idea of the Reich per se to some degree, but they rejected Prussia to a level that could be labeled as pure hate. So the Kings policy led to discontent with monarchy, moreover, social democrats in parliament handed in a plea for parliamentarisation of the state, which met the Kings refusal.

So, just like in the rest of Germany, unrest and  strikes occured in Bavaria, in January 1918. Any attempts to resolve the situation and to save monarchy and monarch came way to late. In November 1918, the radical left, led by Kurt Eisner of the USPD (engl.: Idependent Social Democratic Party of Germany, ISDP), staged a coup in Munich. In this regard, Bavaria was like other states, in which the left did or tried to replace the old system with a new one. The left spread propaganda among soldiers, workers, sometimes farmers and intellectuals to gain their support for their political ideas and beliefs. Usually, the cities and towns would be hotbeds for left wing ideologies, wheread the rural areas tended to be more conservative. This is really nothing too special in comparison to the trest of the Reich. In some states, revolution was violent, in others, it was not.

After their coup, they declared Bavaria to be a Freistaat, hence until today: Freistaat Bayern. As a reminder - please do not confuse that word and the literal translation as Free State with some kind of declaration of independence or as seperatism from the Reich. The term Freistaat in this context means, Bavaria (and other German states who adopted the same term in that time) was a Republik, a republic, from now on.

Anyway - the new government around Eisner relied on Workers-, Citizens-, Soldiers- and, in the countryside, Farmers-Councils. However, the new government lost the elections to the Landtag, the state parliament, while the catholic-conservative BVP (Bavarian Peoples Party) won. Two things happened: Eisner was murdered by Anton Graf von Arco auf Valley, a right wing radical, and social democrat Johannes Hoffmann was elected Prime Minister. It was a hollow victory, since he did not have the majority in parliament, so the left and the radical left clashed. The results were, to make a long story short, the First (more anarchist and pacifist) and Second (communist) Bavarian Soviet Republic.

In suppressing the both Republics lies the root to the second part of the question: Why the radical right. In that way, Bavaria was special.

Witnessing Soviet Republics in a conservative and traditional state actually traumatised many and appealed to a deep rooted rejection of left-wing ideas. Ending both republics was done by Freikorps, Free Corps, and regular army. Thousands of people were sentenced to death or just murdered or sentenced to prison. Red scare was prominent in Bavaria and the right profited from that in the radicalistion that followed.

The new Prime Minister, Gustav Ritter von Kahr (BVP), led a bourgeois-right wing government. He had been a Regional President in Bavaria and witnessed the way that led to the Revolution in 1918, the end of monarchy had crushed his worldview. In his new position, he emphasised on Bavarian autonomous position in Germany and he aimed to change the Free State into a so called Ordnungszelle (Order Cell? Cell of Order? I do not know the proper translation, but both terms get close), meaning a conservative haven in, as (not only) he perceived it, chaotic, jewish and marxist Germany. Kahr relied also on Vigilante Groups and had ties to the National Socialists. During his term, he initiated the expulsion of so-called Eastern Jews in Bavaria as part of a nation anti-semitic campaign. In this mindset, it was only logical to harbor right wing criminals that took part in coups (like the Kapp-Putsch) or were members of violent paramilitary and maybe even alreads terrorist organisation like OC (Organisation Consul). His successor, Eugen von Knilling (also BVP) continued that policy and led Bavaria into conflict with the Reich in 1923, during which the Bavarian Army refused to follow orders from Berlin and von Kahr first was appointed as Commissioner General and then, second, committed high treason.

So, in that regard, Bavaria was special, as mentioned before. For instance, in Berlin there was no development like in Bavaria despite the Spartakus Uprising. After the suppression of both Soviet Republics 1919/1920, the left was oppressed. The right, on the other side, enjoyed more freedoms because they were perceived as useful against the left. Even more so, after Communists and Social Democrats formed governments in Saxonia and Thuringia in 1923 that were legally elected, started to set up Proletarische Hundertschaften (Proletarian Hundreds, kind of a militia) but refused to give in to pressure from Berlin and the Communists used their role as ruling party to prepare the so-called German October, the idea of a German October Revolution. They made good boogeymans for the right in Bavaria, even more so since, all three states share a border.

Selected sources:

Richard J. Evans: The Coming of the Third Reich, New York 2003.

Peter Claus Hartmann: Bayerns Weg in die Gegenwart.Bayerns Weg in die Gegenwart. Vom Stammesherzogtum zum Freistaat heute, München 2012

Martin Hille: Revolutionen und Weltkriege. Bayern 1914 bis 1945, Köln 2018.

Ludwig Hümmert: Bayern, vom Königreich zur Diktatur. 1900 - 1933, Pfaffenhofen 1979.

Michaela Karl: Die Münchener Räterepublik. Porträts einer Revolution, Düsseldorf 2008.

Gabriel Kuhn (ed.): All Power to the Councils! A Documentary History of the German Revolution of 1918–1919, Oakland 2012.

Allan Mitchell: Revolution in Bavaria 1918–1919. The Eisner Regime and the Soviet Republic, Princeton 1965.

Laura Mokrohs: Revolution. Kurt Eisner, Gustav Landauer, Erich Mühsam, Ernst Toller 1918/19 in München, München 2021.

Alois Schmid: Das neue Bayern. Von 1800 bis zur Gegenwart, Bd. 1: Staat und Politik (Handbuch der bayerischen Geschichte, Bd. 4.1), 2te neu bearb. Aufl. München 2003.