Why did Nazis burn 628 Belarusian villages to the ground?

OP: Why did Nazis burn 628 Belarusian villages to the ground?

For unclear reasons there are at least two sets of Wikipedia articles regarding Khatyn (not to be confused with Katyn), which are not interlinked. Thus article Khatyn massacre linked in the OP and the article about The Khatyn national state memorial of the Republic of Belarus provide complimentary information.

The latter article specifies:

The Khatyn national state memorial of the Republic of Belarus is the central war memorial of Belarus for all victims of the German occupation during World War II. It commemorates particularly the more than 600 "burnt villages" which, together with their inhabitants, were destroyed by the Germans throughout the war.

and

The names of 433 Belarusian villages, burned like Khatyn but rebuilt after the war, are immortalized like branches on symbolic "trees of life".

Regarding the reasons for the specific massacre at Khatyn the article states:

The destruction of Khatyn and the murder of the villagers was an act of revenge in response to the bombardment of a German motorcade by Belarusian partisans on 22 March 1943, killing the company commander, Captain Hans Woellke, and three Ukrainian members of the Battalion 118 protection team.

OP: I remember learning that the Nazis thought the Slavic were of a lesser race than the Germans, so they enslaved and mistreated a lot of Poles. Did the Nazis also want to kill them for racial reasons?

Nazis considered Slavs as an inferior race, but not as a race subject to the immediate extermination (as they treated the Jews.) Thus, the treatment of the enemy combatants, prisoners of war, and civilians was strikingly different in the western and eastern fronts - e.g., milliosn of Soviet prisoners of war were simply allowed to starve (see German atrocities committed against Soviet prisoners of war.) Likewise, although Germans did commit atrocities in the west, particularly towards the end of the war (see, e.g., Oradour-sur-Glane massacre), such reprisals were a systematic policy in the east from the first days of war. The already cited Wikipedia article states:

Of the total number of 5295 villages destroyed, 3% were destroyed in the first year of war in 1941, 16% in 1942, 63% in 1943 and 18% in 1944.

Nazi plans were not entirely consistent.

On the one hand, they wanted Lebensraum in the East for Germanic settlers. That was the very-long-term vision. On the other hand, they wanted the slave populations to work plantations. That was a vision for decades. And finally, they wanted workers for the (war and agricultural) industry in Germany, if possible without dependents to divert their attention. That was on the short term. On the very short term, they also recruited non-Jewish populations in the East to kill Jews and partisans, killing some who did not volunteer to encourage the others.

Nazis were not primarily in control of German racial policy in the East. The heer was primarily in control. After the heer, security apparatus controlled ultimately by the SS organisation, but often proximately by former police or former heer (for example in Reserve Police Battalions), designed and implemented racial policy. Finally various factions, vibes and movements inside the Nazi party gesticulated various racial plans. These confused plans could exist separately because of the administrative concept of working towards the Fuhrer: organisations would have separate and overlapping policy controls so long as they pre-emptively anticipated what the central leadership wanted or could want.

Village massacres were predominantly planned by security apparatus including the highly Nazi "Einsatzgruppen" or "Action-Groups" of racial political police, whose primary police action was organising genocidal massacres of population centres. These Action Groups called on the assistance of Reserve Police Battalions with more general anti-partisan and racial duties, and on the assistance of Heer units in rear areas. While the Reserve Police Battalions were openly political in nature, they were not exclusively or predominantly genocidal: their main political activity was massacring civilians on the basis of political or military resistance, not due to racial extermination. Heer units involvement was always a matter for local officers with policy control: for commanding officers the decision to allow heer units to participate in racial massacre was a voluntary one. It was usually the case that the Heer provided assistance. It was not, however, universally the case. More interesting still is that neither Reserve police battalion nor heer troops who refused to engage in massacre were penalised, they were given other light relaxing duties such as preparing food for genocides. The seminal text here is Browning, Christopher (1993) Ordinary men: reserve police battalion 101.

Most of these massacres were not by burning, but by shooting people into ditches. These massacres were based on a belief that slavic and other soviet peoples were racially different to the people making up German military forces in the East, and that slavic people were deserving of death. Racial massacre seems to have been lacking in systematisation, and does blend into anti-communist political massacre. The chief way to differentiate these, both of which were generally called "anti-partisan" operations, is that one was actively initiated, and the other nominally reprisal.

The chief other instance of German racial policy in the East was developed by the Heer who systematically starved captured soviet troops not afforded standard prisoner of war status or protection. This starvation happened primarily over 1941/2, and was connected both to the idea that slavic and other soviet peoples were racially other and deserving of death, and that the Heer has a responsibility to prevent the humanitarian, racial and political impact of a 1917 food deficiency in Germany on the German people by deliberately starving other peoples. 1917 was an animating imaginary for right wing members of the Heer as it was viewed as starting the communist elements of the German revolution, and defiling the imaginary German nation.

In summary:

German policy was not centralised, but developed in multiple centres in anticipation of what administrative units believed that the Nazi leadership wanted, and in line with those units interests.

Chief responsibility for German policies of genocides against Soviet Citizens relate to the implementation of administratively local Jewish Policies and Slavic Policies (this is slightly confused as both were considered partly coterminous with "communism" which was considered racially). The chief organisations responsible for these policies were the German Army, and the security forces which had been merged into the SS and which varied from Nazi dominated as in the Einsatzgruppen to old men of mixed political background if slightly less communist than normal and slightly more petit-bourgeois than normal organised by pre-Nazi police leaderships.

The chief method of extermination of Soviet Citizens other than Jewish Soviet Citizens was deliberate starvation, followed by localised massacre. Burning villages with their inhabitants was atypical and inventive compared to the ordinary method of making the villagers dig a ditch and then shooting them into it.

Browning C (1993) Ordinary men. This seminal text is the grounding of literature, and exemplifies the functionalist and massacre specific bent that historiography has taken since its publication.

Another answer has cited the organisatorial level of the eliminatory actions of German forces in the Soviet Union from Christopher Brownings seminal book. I'd like to add the arguments offered by the regime and the military leadership for these actions.

Essentially, the fight against partisans, the racial prerogative of reducing the Slavic population, or at least its subdueing by means of terror, and the elimination of the jews are all mixed up in a big bowel of distrust, hatred and cruelty.

The following is cited from Reinhard Rürup (Hrsg.), Der Krieg gegen die Sowjetunion 1941–1945. Eine Dokumentation zum 50. Jahrestag des Überfalls auf die Sowjetunion. (Begleitband zur Ausstellung in der provisorischen Halle der "Topographie des Terrors".) Berlin 1991. Abschnitt: Die "Bandenbekämpfung", S. 127–139. Translations with the help of DeepL.

In der deutschen Kriegsführung gegen die Sowjetunion wurden von Anfang an elementare Grundsätze des Völkerrechts mißachtet. Auf andere Weise nahm allerdings auch der Aufruf Stalins am 3.7.1941 zum Partisanenkrieg hinter der deutschen Front keine Rücksicht auf traditionelle Regeln der Kriegsführung.

Das Interesse der Wehrmacht an der Sicherheit der Truppen, an der Sicherung der zentralen Versorgungs- und Nachrichtenverbindungen und an der umfassenden wirtschaftlichen Ausbeutung der besetzten Gebiete verschränkte sich in der Praxis der sogenannten „Bandenbekämpfung” mit dem rassenideologischen Herrschaftskalkül von SS und Polizei, aber auch von Teilen der Militär- und Zivilverwaltung, einer „Dezimierung” der einheimischen slawischen Bevölkerung. Schon seit Juni 1941 bildeten die „Befriedungsaktionen” in den eroberten Gebieten auch einen wesentlichen Bestandteil des systematischen Massenmordes an den Juden. Der Höhere SS- und Polizeiführer Rußland-Mitte und spätere „Bevollmächtigte für Bandenbekämpfung”, Erich von dem Bach-Zelewski, prägte im September 1941 den Satz: „Wo der Partisan ist, ist der Jude, und wo der Jude ist, ist der Partisan.”

In the German warfare against the Soviet Union elementary principles of international law were disregarded from the very beginning. In another way, however, Stalin's call on July 3, 1941 for guerrilla warfare behind the German front did also not take into account traditional rules of warfare.

The Wehrmacht's interest in the security of the troops, in securing the central supply and intelligence links and in the comprehensive economic exploitation of the occupied territories intertwined in the practice of the so-called "fight against bandits" with the racial ideological domination calculation of the SS and the police, but also of parts of the military and civil administration, of a "decimation" of the native Slavic population. Already since June 1941, the "pacification actions" in the conquered territories also formed an essential part of the systematic mass murder of the Jews. The higher SS and police leader of central Russia and later "Procurator General for Bandit Combat," Erich von dem Bach-Zelewski, coined the phrase in September 1941: "Where the partisan is, there is the Jew, and where the Jew is, there is the partisan."

A meeting of the Nazi leadership on Juli 16th, 1941 was recorded by Martin Borman in the following way:

[Einleitende Bemerkungen des Führers]
Die Russen haben jetzt den Befehl zum Partisanen-Krieg hinter unserer Front gegeben. Dieser Partisanenkrieg hat auch seinen Vorteil: er gibt uns die Möglichkeit, auszurotten, was sich gegen uns stellt. [...]
Reichsleiter Rosenberg schnitt dann die Frage der Sicherung der Verwaltung an. Der Führer sagt [...]
Der Riesenraum müsse natürlich so schnell wie möglich befriedet werden; dies geschehe am besten dadurch, daß man Jeden, der schief schaue, totschieße.
Feldmarschall Keitel betont, für ihre Dinge müsse man die Einwohner selbst verantwortlich machen, denn es sei natürlich nicht möglich, für jeden Schuppen und für jeden Bahnhof eine Wache zu stellen. Die Einwohner müßten wissen, daß jeder erschossen würde, der nicht funktioniere, und daß sie für jedes Vergehen haftbar gemacht würden.

[Opening remarks of the Führer]
The Russians have now given the order for partisan warfare behind our front. This partisan warfare also has its advantage: it gives us the opportunity to eradicate what opposes us. [...]
Reichsleiter Rosenberg then touched on the question of securing the administration. The Führer says [...]
The giant area must of course be pacified as quickly as possible; this is best done by shooting to death anyone who looks askance.
Field Marshal Keitel emphasizes that the inhabitants themselves must be held responsible for their affairs, because it is of course not possible to provide a guard for every hovel and for every railway station. The inhabitants had to know that anyone who did not work would be shot and that they would be held liable for any offense.

A command by General Field Marshal von Reichenau, leader of the 6th Army, from November 11th, 1941 orders the following:

Soldaten der 6. Armee! Ihre habt als Rächer anzutreten zum organisierten Kampf gegen die gewissenlosen Mordbestien! Zweierlei ist dazu nötig. [...] zweitens müßt ihr Mittel zur Vernichtung dieser Mörder anwenden, die weder unserer Art entsprechen, noch jemals von deutschen Soldaten gegen eine feindliche Bevölkerung angewendet worden sind.
Dazu befehle ich: [...]
2.) Alle Dörfer und Gehöfte, in denen Partisanen beherbergt und verpflegt wurden, sind durch Einziehen der Lebensmittel, Abbrennen der Häuser, Erschießen von Geiseln und Aufhängen der Mitschuldigen zur Rechenschaft zu ziehen, wenn nicht einwandfrei nachgewiesen wird, daß die Bevölkerung sich selbst gegen die Partisanen gewehrt und dabei Verluste erlitten hat. [...]
Die Furcht vor unseren Vergeltungsmaßnahmen muß bei der Bevölkerung größer sein als die vor den Partisanen.

Soldiers of the 6th Army! You have to stand up as avengers for the organized fight against the unscrupulous murder beasts! Two things are necessary for this. [...] Secondly, you must use means for the destruction of these murderers which neither correspond to our kind nor have ever been used by German soldiers against an enemy population.
For this I order: [...]
2.) All villages and homesteads in which partisans were harbored and fed are to be held accountable by confiscating food, burning the houses, shooting hostages and hanging the accomplices, unless it is proven beyond doubt that the population itself fought the partisans and suffered casualties in the process. [...]
The fear of our reprisals must be greater among the population than that of the partisans.

A special role in Belarus was played by SS and police leader in "Weißruthenien" Curt von Gottberg. He basically accused anyone he encountered to be an enemy. Wikipedia cites:

Gottberg developed a new 'strategy' in the fight against partisans on the occupied territory of the Soviet Union, mounting aggressive operations against suspected 'partisan bases' (generally ordinary villages; Gottberg's strategy seems to have largely involved terrorising the civilian population). Whole regions were classified as "bandit territory" (German: Bandengebiet): residents were expelled or murdered and dwellings destroyed. "In the evacuated areas," said Gottberg in an order, "all people are in future fair game". An order of Gottberg's of 7 December 1942 stated: "Each bandit, Jew, gypsy, is to be regarded as an enemy".

His own battle group, together with the Sonderkommando Dirlewanger and the Kaminski-Brigade is responsible for the organised mass murder of countless civilians in Belarus, especially in the operations

• Nürnberg (report from 5/12/1942): "Enemy dead: 799 bandits, over 300 suspected bandits and over 1800 Jews",
• Erntefest I (18/1 to 26/1/1943): 805 dead enemies, 1.165 "Sonderbehandelte" (murdered) and
• Erntefest II (28/1 to 9/2/1943): 2.325 dead enemies

The Sonderkommando Dirlewanger is described in detail in Christian Ingrao: Les chasseurs noirs. La brigade Dirlewanger. Perrin, Paris 2006.