This is exactly one of the questions the Germans themselves wondered in the 19th century. Until the middle of the 19th century it was quite common to write teutsch and Goethe was praised in the dictionary of the brothers Grimm for using deutsch ("the right form").

That it stems from the Teutones or Teutoni (a roving tribe of either Celts or pre-Germans which the Romans defeated in 112 B.C. during the Cimbrian war, their leader Teutobod was strangled after the Triumph of Gaius Marius, what happened to the remainder of the host wasn't recorded) is one of the popular explanations.

However, the first definitive recordings of deutsch in the context of East Francia are theodiscus in clerical scripts regarding the use of the local germanic language in East Francia, as opposed to Latin, during the 8th century. It comes from a germanic word, thiota or thiot (the Name Dietrich comes from that word), meaning "the people", theodiscus meant the language the local people spoke.

Over the years, the inhabitants of East Francia became known as theodiscus/ theodisci. Starting in the 11th century, this word is replaced with teutonicus/ teutoni which was taken to mean the same. How much this was inspired by the older Latin usage is not known.

Maybe the first mention of Deutschland is "in Diutischemi lande" in the Annolied of about 1080 meaning the lands of the Bavarians, the Saxons and the (East-)Franks. It seems, however that this took a lot of time to catch on and only is used sparely during the Middle Ages as in Diutisk lant or variants thereof. The early modern period sees the common usage of Tiutschland from the 16th century onwards.

The form deutsch is used by Martin Luther, who had tremendous influence on the developement of Hochdeutsch (the modern high language), which is maybe the reason the Grimms took it for "the right form".

Source: Digitales Wörterbuch der Deutschen Sprache, Berlin-Brandenburgsche Akademie der Wissenschaften

Short answer: "Deutsch" is the term by which Germans have been distinguishing themselves since the early middle ages.

Long answer:

"Thiudisk" was originally an adjective for "of the people", describing the language used by the Germanic speaking East Franks. The oldest remaining document using it is the translation of the Bible into Gothic by Bishop Wulfila, where the term is used to translate "Gentiles" in Galitians 2:14: "(...) jabai thu Iudaius wisands thiudisko libais jah ni iudaiwisko, hvaiwa thiudos baideis iudaiwiskon?" (If thou, being a Jew, livest after the manner of Gentiles, and not as do the Jews, why compellest thou the Gentiles to live as do the Jews?)

The term was latinized into "theodiscus" and can be found in numerous sources starting from the late 8th century. For example, in the account of the Carolingian imperial assembly of 788, the present Bavarian, Langobardian and Saxon representatives where summarized using this term. This originally cultural distinction became a political one with the split of the Carolingian empire into three parts by the Treaty of Verdun (843). However, it took until the 12th century for the term "diutisk-lant" to be used in the sense of the country in the so-called "Kaiserchronik" (Imperal Chronicle): "do er die heiden alle betwanc, do vuor er in diutisk lant" (As he conquered the Pagans, he went to Germany). The legend of Pilatus, written around the same time, mentions a "tutisch volk", i.e. a German people.

Finally, the oldest existing mention of the term to describe a nation can be found in a written Pledge by a count Hermann from 1474: "im heiligen romischen rych der duytschen nacioin ..." (in the Holy Roman Empire of the German Nation...). From early 16th century on, the term was used until the end of the Holy Empire with the abdication of Francis II in 1806.

Between that time, "Teutschland" or "Deutschland" was generally used to mean all German-speaking countries, e.g. by Martin Luther, in a book of prohesies from 1569 ("Weissagung Simeonis, ober itziger zeit Deutschland"), in a chronicle of the 30 years war from 1650 ("Summarische Chronick Oder eigentliche Verfassung deß ein und Dreissig-jährigen von Anno 1618 in Teutschland geführten Kriegs"), in the writings of Friedrich Nicolai in the 18th century like "Briefe über den itzigen Zustand der schönen Wissenschaften in Deutschland" etc.

The unification of "Deutschland" into a single country rather than the loose "Deutscher Bund" after the Congress of Vienna (1814) remained the dream of many. That dream was expressed e.g. at the Hambach Festival (1832) as well as in the German Revolutions of 1848.

So, when Bismarck united the German countries with the exception of Austria, the term "Deutsches Reich" must have come rather naturally.

The word seems related to the Teutones of Classical times - the "-on" is a common Greek ending, suggesting a root like "Teut", from which Teutsch ("Teut-ish", we might say), and later Deutsch. Why their name particularly should be so perpetuated (as in "Teutonic") out of all the early Germanic peoples' is unclear, but "Deutsch" had been used for the German people for many centuries.

English has a related name for a Germanic people, but closer neighbours got to claim it first - duits -> Dutch.