I think you already know what "thee, thou, etc." translates into german. And you are right: there are no exactly corresponding words which have this old touch in german language (maybe there are, but i can't think of any right now and Bethoven doesn't use any in his Testament). It's correct to translate them like fragezeichen & PiedPiper did, but i don't think that was your question. Your question was why the translators used these and other old Elizabethan English words.
Well, have a look at the first sentence of the "Heiligenstadt Testament"
O ihr Menschen die ihr mich für Feindseelig störisch oder
Misantropisch haltet oder erkläret, wie unrecht thut ihr mir, ihr wißt
nicht die geheime ursache von dem, was euch so scheinet, mein Herz und
mein Sinn waren von Kindheit an für das zarte Gefühl des Wohlwollens,
selbst große Handlungen zu verrichten dazu war ich immer aufgelegt,
aber bedenket nur daß seit 6 Jahren ein heilloser Zustand mich
befallen, durch unvernünftige Ärzte verschlimmert, von Jahr zu Jahr
in der Hofnung gebessert zu werden, betrogen, endlich zu dem überblick
eines daurenden Übels
When you translate a text you can translate its meaning and message and thats it. But you could also translate its flow and its tone and in that case you have to look at the words and grammar the author used (especially in old texts). In this case its a very antiquated text, so you use antiquated english words: ergo Elizabethan English
This text is old and it sounds very very old. No one would talk like this today. No one would write a lot of words like this today. Words like "erkläret" or "scheinet" sound outdated so i think it is legit to use words like "thou", "lookest" or "understandest" which have the same old flow, when translating this text while trying to retain its style.