super-tl;dr - around the time of Martin Luther, sixteenth century

tl;dr - With good knowledge of modern German, you would be able to recognize thirteenth-century Middle High German texts as German and grasp chunks, but you might struggle to put together the overall picture depending on the type of text (narratives are easier). You would be able to read through early fifteenth-century MHG texts by highly educated authors without too much difficulty, but would struggle mightily (even with a dictionary of local terms) with poems and letters by less-educated lay people. In the early sixteenth century, a few words in Martin Luther's Early Modern High German might trip you up, and you'd definitely notice some spelling differences and word order fluidity. But while the people writing letters to Luther might sometimes be a bit harder to understand, you'd feel like you were reading fluently at this time. Finally, a series of orthographic conferences around the turn of the twentieth century ushered in some of the spelling standardizations that you know today.

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Languages evolve over time; you can't take two texts written in subsequent years and say, "Aha! THIS is where Old English ends and Middle English begins." So although linguists and historians will talk about different stages in the transformation of a language, there is a lot of flux. Additionally, there was much more regional and educational-level variation in vernacular (non-Latin) languages across Europe before the printing press and the need for people in one town to understand texts purchased in or borrowed from another gradually helped "fix" one dialect as standard. With all those disclaimers in mind:

The language we know as German evolved out of the West Germanic language group that also includes the ancestors of English, Low German and Dutch. This pre-written language would have been spoken around the final years of the Roman Empire, through the fifth or sixth century AD. At this stage, the ancestors of German and English are very close, so you can picture one step more primitive than Beowulf for an idea of what this era of German might have sounded like. By the time of the earliest full texts written in German (there are some earlier brief inscriptions), 8th-9th century or around the time of Charlemagne, the regional differences in the West Germanic languages have deepened to the point scholars mark out Old High German, Old Saxon, and Old English (Anglo-Saxon), but regional dialects ebb and flow into each other. I know how we feel about Wikipedia around here, but Wiki's illustration of the dialect difference with the Lord's Prayer illustrates really nicely both the importance of dialect and how far we are at this point from a language immediately legible to modern German speakers. (Here's the modern version.)

Although the few people who are writing around the turn of the millennium are doing it mostly in Latin, a handful of German texts survive that point to a simplification in spelling: unstressed vowels get switched to 'e'. So all those -en infinitive verbs and -e, -en plurals? That's when that shows up. Scholars figure this development is as good as any point to say that OHG has evolved into Middle High German.

The later twelfth and thirteenth centuries are when the vernacular languages in western Europe, including German, start to attract authors (especially women, who have less access to Latin education than men) and flourish. MHG grammar, such as pronouns, noun/adjective declension, and verb conjugation in a lot of ways is recognizable to modern German speakers, but there is a LOT of different vocabulary. Spelling differences and an utter lack of spelling standardization also mean that someone coming to a medieval text with no MHG experience will be utterly reliant on a dictionary, but can probably muddle through thanks to the grammar similarities as long as you know a few key tips like extra negatives make the negative emphasis stronger, they don't cancel it out like an English double negative. See what you can make out of the MHG version of Mechthild of Magdeburg's Flowing Light of the Godhead, a mystical text often viewed as foundational in the development of written German.

Where MHG ends and Early Modern High German begins is kind of fluid. The most educated clerics writing in dialects close to Central German/Saxon are probably recognizable as EMHG around the time of the Black Death, ~1350. Basically, some weird MHG diphthongs (uo, iu) have started to go out of fashion, and the spelling and vocabulary feel more comfortable in general. In the fifteenth century, to me at least there is a striking difference between reading highly educated people writing in a Saxon-esque dialect, versus works by less educated lay people in more southern/western dialects (Bavarian, Alemannisch). It's the difference between reading, and reading through cobwebs.

In the early sixteenth century, Martin Luther and his Bible translation traditionally get a lot of credit for their role in standardizing EMHG around one dialect. Modern scholarship, however, generally accepts that Luther specifically chose the Saxon dialect to work in because he thought it was the most "average", comprehensible to most High German speakers, and already gaining ground as a common choice of printers. (In England, Chaucer's relationship with the standardization of Middle English is similar, albeit occurring earlier). You can see in, for example, Luther's translation of the Gospel of John ("In the beginning was the Word, and the Word was with God") that we've arrived at a language reasonably close to modern German and easily comprehensible, with some fairly standard spelling differences.

A good hallmark of the evolution of medieval into early modern version of languages is probably that modern publishers find a market for translations of Middle English and Middle German texts into the modern form of the language (Chaucer's Canterbury Tales, Wolfram's Parzival). You are most unlikely to find a massive demand for modern German versions of Luther's letters or an English update of Anne Askew's Examinations.

As to the 19th-20th century standardization efforts and spelling reforms, I'll defer to the modernists here.

Sources: Honestly, I don't even know at this point. A lot of this is firsthand observation from working with MHG/EMHG texts, reinforced with the introductions to countless linguistics books/dictionaries while trying to track down an obscure word or its possible roots. A solid English overview of the history of German, and a cool way to go about seeing the changes for yourself, is Thomas Gloning and Christopher Young, A History of the German Language Through Texts.

ETA: To clarify on "you are not a source" grounds--if necessary, I'm happy to provide documentation to footnotes expressing other scholars' frustration with the thick dialects and, shall we say, lack of linguistic sophistication in the texts I'm basing my "reading through cobwebs" comments on. I hope the note about modern translations versus simple editions also serves as evidence to back up personal anecdotes.