Several reasons.

First, Theodoric was King of Italy thanks to an agreement with the Byzantine Roman Emperor Zeno, and as such, his kingship gave him reach well beyond any linguistic differences that divided Ostrogoth from Visigoth (or even Ostrogoth from Frank, Vandal or Burgundian, some of the other recognizable Germanic groups kicking around in the late Antique period).

Two, his ruling policy included diplomacy across a very wide geographic reach and marriage alliances (his own or his kids) with several other elite members of these Germanic groups.

Third, he actively took advantage of Roman culture and Roman Imperial norms, including the retention of a secretary, Cassiodorus, whose letters positioned Theodoric as a sophisticated Roman with full control of Classical Latin epistolary style. This last bit also means Theoderic’s rule was preserved in Latin, which crossed pan-European linguistic lines for anyone educated well enough to read it between the end of Theodoric’s reign in the 520s and the 10th/11th century beginnings of Middle High German sagas that include Dietrich.

Bryan Ward-Perkins has a good general overview of Late Antiquity in The Fall Of Rome (2005) that puts Theodoric’s reign in good context, and you can read some of “Theodoric’s” diplomatic correspondence ’ letters in Barnish’s abridged translation of Cassiodorus’ Variae.

Source: am medieval history professor.

Roman artifacts start showing up in Scandinavia and north Germany as far back as the first century BC which is the start of the 'Roman Iron Age' in the area, defined by the preponderance of such artifacts and the cultural changes caused by Roman influence. No small part of this influence was caused by Germans serving as auxiliaries/mercenaries in the Roman armies. A notable influence on Germanic culture beyond artifacts was the creation of the runic writing system . It had come into existence by the second century CE and was an influence of Old Italic and/or Latin alphabets. A number of Latin loans also exist already in Proto-Germanic, including the word for 'to write' (*skrībaną). By Theodoric's time, there are Byzantine artifacts in Scandinavia as well.

So you cannot assume there was 'no cultural contact' between Germany and either Italic/Romance or Gothic culture in Italy. Northern Europeans travelled - even before the Vikings. It's known there was contact, and sufficient contact that language wouldn't necessarily be an issue even if Ostrogothic wasn't immediately intelligible. (you don't end up with loanwords unless someone knows a bit of both languages) In any case and perhaps more significantly, you have the Lombard invasion not long after Theoderic, and Lombardic certainly mutually intelligible with what was being spoken in Germany since their conquest of Italy was not even a century after leaving.

Theoderic's legend literally preceded written history. He is mentioned on the 9th century Rök runestone in Sweden, which is probably the most famous of all runestones besides also being one of the longest, most enigmatic and most important. There are dozens (over 50 published in fact) interpretations of the stone, and much isn't understood but the Theoderic reference is fairly settled. The fact that he's described as being on a horse has generally been taken to refer to the equestrian statue of Theoderic which Charlemagne had moved from Ravenna to his palace in Aachen - so that too is another possible influence. The identification of the Goths with Scandinavia and the Geats goes back to Jordanes writing as much in Getica centuries before this.

Which does not, however, mean that the 13th century Scandianvian Thidrekssaga forms a continuous tradition with the Rök Stone. It does not, it came from a Low German source. (and have prefaces claiming as much)

The actual story of Dietrich von Bern is not the oldest Germanic written narrative about Theoderic either thoug, the Hildebrandslied is older, and the oldest manuscript of that is about contemporary with the Rök stone. So an oral tradition preceded any written accounts we have.