How did three characters for _de_ emerge?
The premise of the question is a bit backwards. It's not that de evolved into three different characters, it's that three different words evolved to have the same pronunciation in modern Mandarin Chinese.

Mandarin in particular, features unstressed syllables, which are commonly referred to as having a "neutral tone" rather than having one of the four main tones. Most of the Chinese languages are considered syllable-timed languages, including Mandarin, but it is less so compared to other Chinese varieties such as Cantonese, in which all syllables are stressed and have a tone. Syllable-timed languages typically allocate the same amount of time between all syllables in their speech rhythm. In contrast, stress-timed languages (such as English) allocate the same amount of time between stressed syllables. Mandarin seems to have some aspects of stress-timing in which neutral tone syllables are pronounced with shorter durations. As is typical with stress-timed languages, unstressed syllables often undergo vowel reduction, especially in fast speech.

This vowel reduction is one of the main contributors to how all of these words ended up being pronounced with a neutral de in Mandarin. As fefe mentioned in this comment, they are not necessarily pronounced the same in other dialects (for instance, 得、地、的 are respectively pronounced dāk, déi, and dīk in Cantonese). Here's how each of these characters came to be pronounced de in modern Mandarin:


得 - derived from its original meaning of "to attain". One of the features of Chinese is serial verb construction, in which multiple verbs can be stacked together. It is through serial verb construction that 得 evolved into its modern usage as a verb complement to indicate the degree to which an action was attained.


  跑得快 - "run and attain quickness" → "run quickly"


The pronunciation of this word in Middle Chinese (the Chinese of ca. 600 - 1000 AD from which most of the modern Chinese varieties evolved) had a -k ending, but the Mandarin dialects evolved to drop -k endings from all syllables, leading to its modern pronunciation of dé (for comparison, this word is pronounced dāk in modern Cantonese). When used as a verbal degree marker, it becomes unstressed, resulting in its neutral de pronunciation.

Also note that while the stressed dé and unstressed de appear to have the same pinyin spellings besides the tone, the vowel quality is different. Expressing this in IPA, the stressed dé is pronounced [tɤ] with a close-mid back unrounded vowel, while the vowel in the unstressed de has reduced to a schwa: [tə]
地 - the etymology of its use as an adverb marker is less clear, but this suggests that it's related to its original meaning of "place":


  Ōta (2003) proposes that it is a grammaticalization of etymology 1. The construction ADJ + 地 would have semantically developed as follows: "a ADJ place" → "in a ADJ place" → "in a ADJ manner".


The pronunciation of this word in Mandarin, dì, is very similar to its pronunciation in Middle Chinese. It also became unstressed when used as an adverb marker, and through vowel reduction, ended up being pronounced as de.
的 - the use of this as a genitive/possessive particle is actually unrelated to the original usage of 的 as a word meaning "target". It's most likely instead derived from 之, which is the genitive particle of Classical Chinese. While the pronunciation of 之 evolved to become zhī in modern Mandarin, it had also retained a second pronunciation that was closer to its original pronunciation in Old Chinese, dī.  The character 的 was later co-opted to represent this usage when it was pronounced as dī (historically 底 had also been used before modern written Chinese standardized on 的). Again, through destressing and vowel reduction, it also ended up being pronounced as de.

The previous answers here also provide more details about the evolution of the use of 的 as a genitive particle:


Michaelyus' answer to "Do 之 and 的 come from the same word?"
Betty's answer to "Why is 的 (de) sometimes pronounced “di” even though it is used as a possessive particle?"


As an aside, the use of 的 in its original meaning of "target" was also originally pronounced with a -k ending, which is why modern Cantonese pronounces this word as dīk, even though its modern borrowing for use as a genitive particle is not related to this original word and did not have a -k ending. Its use as a genitive particle is not a native use in Cantonese though, so Cantonese ended up applying the original dīk reading of 的 to its genitive meaning too.