What is meant by "state people" in this context? (Governments of new Balkan states post-WWI)

In the original "state people" is Staatsvolk or the national people of a nationalist state. Arendt's usage is somewhat in contrast to the definition of that Wikipedia link:

The people of a state are usually understood to be the sum of citizens who have a legal relationship with their state and of persons who may in principle be on an equal footing with them under constitutional law but this does not refer to a people in the actual ethnic sense or part of a people living in a state (ethnic group); rather, it refers to people with common citizenship, i.e. citizens of a state (citizens), irrespective of the nationality (ethnicity, origin) of the individual citizen. As a society, a special personal relationship to the state is established for citizens in addition to their regular subjection to state power (at least when staying in Germany): citizenship is a status that establishes reciprocal rights (at least in democracies) and duties for citizens.

Arendt criticises that in Versailles the organising principle was claimed to be nationalism, which was incompatible with actual settlement patterns on the ground in middle and Eastern Europe, where people, peoples and 'nations' lived side-by-side and intermixed. That is meant with "declared to be", as they were usually just one of several subgroups.

For a closer look at the example Czechoslovakia:

The new country was a multi-ethnic state. The population consisted of Czechs (51%), Slovaks (16%), Germans (22%), Hungarians (5%) and Rusyns (4%). Many of the Germans, Hungarians, Ruthenians and Poles and some Slovaks, felt oppressed because the political elite did not generally allow political autonomy for minority ethnic groups. This policy led to unrest among the non-Czech population, particularly in German-speaking Sudetenland, which initially had proclaimed itself part of the Republic of German-Austria in accordance with the self-determination prinicple.

There Czechs and Slovaks were the "state peoples" by name and treated as such by the other powers. Representatives of the other minorities were simply ignored when they complained about their grievances. "The Czechs should have their own state" was a goal, but one that going by strict nationalism would have been utterly impossible to construct boundaries and borders for.

More details to be found when looking for Treaty of Saint-Germain-en-Laye (1919).

The quote again in more context:

The inadequacy of the Peace Treaties has often been explained by the fact that the peacemakers belonged to a generation formed by experiences in the pre-war era, so that they never quite realized the full impact of the war whose peace they had to conclude. There is no better proof of this than their attempt to regulate the nationality problem in Eastern and Southern Europe through the establishment of nation-states and the introduction of minority treaties. If the wisdom of the extension of a form of government which even in countries with old and settled national tradition could not handle the new problems of world politics had become questionable, it was even more doubtful whether it could be imported into an area which lacked the very conditions for the rise of nation-states: h*m*geneity of population and rootedness in the soil. But to assume that nation-states could be established by the methods of the Peace Treaties was simply preposterous. Indeed: "One glance at the demographic map of Europe should be sufficient to show that the nation-state principle cannot be introduced into Eastern Europe." The Treaties lumped together many peoples in single states, called some of them "state people" and entrusted them with the government, silently assumed that others (such as the Slovaks in Czechoslovakia, or the Croats and Slovenes in Yugoslavia) were equal partners in the government, which of course they were not, and with equal arbitrariness created out of the remnant a third group of nationalities called "minorities," thereby adding to the many burdens of the new states the trouble of observing special regulations for part of the population. The result was that those peoples to whom states were not conceded, no matter whether they were official minorities or only nationalities, considered the Treaties an arbitrary game which handed out rule to some and servitude to others. The newly created states, on the other hand, which were promised equal status in national sovereignty with the Western nations, regarded the Minority Treaties as an open breach of promise and discrimination because only new states, and not even defeated Germany, were bound to them.
_{(Hannah Arendt: "The Origins of Totalitarianism", Harcourt Brace & Company: San Diego, New York, London, 1976, p270.)}