According to Bruce Frantzis (a well known Tai Chi Master) in an appendix section of his book the Power of Internal Martial Arts and Chi on the history of Tai Chi, the answer is as follows (it's quite long so this is a paraphrased version, for the longer one you need to read the book).

Tai Chi's history can be broken into 2 parts:

Pre and Post Chen Village.

The Pre Chen village is covered in obscurity and speculation, the post Chen Village timeline is quite clear.

Since all current high level Tai Chi in China came from the Chen Village, I'll discuss the simple one first.

The Chen Village possessed a phenomenally powerful martial art style which allowed them to prosper. For reasons (presumably to maintain their village's status), this art was kept secret and outsiders were forbidden from learning or entering the village. A young man named Yang Lu Chan was an avid martial artist student, who eventually reached the point where his teacher felt he could teach him no more, and recommended him going to the Chen Village to learn Tai Chi.

Being extremely motivated, Yang pretended to be deaf and was accepted as a servant eventually becoming trusted enough to gain the keys to the inner training area. In the evening when current master Chen Hsing was training his students, Yang would observe the class from afar listening intently then once it ended practice the material late into the night before waking up early in the morning to begin his normal job as a "deaf" servant.

After 6 years, Yang was caught, with many of the Chen village members wishing to throw him out or kill him on the spot. Yang managed to be given the opportunity to demonstrate his abilities, and was able to easily defeat each other student, arising Chen Hsing's curiosity. This in turn led to him testing Yang for suitability as a future disciple (as Yang has already demonstrated extraordinary perseverance and discipline), and once he passed Yang spent the next 18 years training intensely, at which point Chen Hsing gave Yang Lu Chan his blessings to spread it to the outside world.

Once left the Chen Village, his martial skills were at a phenomenally high level and he was able to defeat every opponent, or groups of attackers that came at him with ease, leading to him becoming known well enough to be recognized by the Emperor as the top martial artist in China and get the job that came along with this distinction. Yang also taught regular citizens, and his top disciples passed on the art allowing it to spread throughout China through the ages, primarily with Yang's style and one originating from a top disciple, the Wu Style one designed with an emphasis on healing (a third style called the Hao style also exists).

After Tai Chi was introduced to the Chen village, they spent over a 100 years developing and perfecting the art but how it first arrived there is a matter of speculation. 4 main theories exist to explain the pre chen period, each of which is often cited as the authoritative source for Tai Chi's history but no reliable evidence proves one over the others.

Theory 1

A famous taoist immortal (one who had achieved true enlightenment) named Chang San Feng watched a snake and a crane fight and from this deduced a new martial arts style which differed from those of the era, which eventually was passed down to Wang Tsung Yueh who randomly arrived one day in the Chen Village (an odd occurrence since it was in the middle of no where and far away from the Shaolin Temple).

[pause theory 1]

Theory 2

Wang Tsung Yeah learned tai chi from one of many existing Tai Chi lineages (possibly Chang San Feng or even an older one) and randomly arrived at the local inn [resume theory 1] full of closed Chen Family insiders who held a great deal of pride in their village's martial arts. Wang disparaged and then insulted the village's martial arts, which provoked them into attacking Wang only to be completely defeated. In awe of his abilities, they begged him to teach them-but lacking the time to fully teach the art he instead adapted its energetic components into their Shao Lin Cannon Fist style.

Theory 3

Tai Chi was created by a resident of the Chen village, Chen Wan Ting who was a top general in the Chi Chi Guang army. Chen created Tai Chi through combining elements of Chinese Medicine (especially the meridian line theory) with Chi Chi Guang's Shaolin martial art methods. This theory is largely supported by the overlaps between Chi Chi Guang's military manual and the basic techniques in the Chen Style.

Theory 4

Chen Wang Ting (still a general but belonging to a different army) gave Jiang Fa political sanctuary in the Chen village in return for him teaching Chen's clan Tai Chi Chuan. Jiang Fe happened to know both Wang Tsuang Yeah's energetic power system and Chi Chi Guang's martial arts methods (which were basically a compendium of the best existing Shao Lin techniques). It is not known how Jiang Fe came to possess this knowledge, but after he imparted it to Chen Wa Ting, creating the First Set of Chen Style Tai Chi, Chen Wa Ting then later developed the second half (Cannon Fist) on his own.

Although Bruce believes the history is too vague to ever be certain of the exact origin of Chen Tai Chi two facts are clear.

a) Chen Tai Chi partly derived from Chi Chi Guang, there are too many similarities between the two.

b) Chen Tai Chi's energetic components were too advanced to have just been created through an adaption of meridian line theory (they are too simple).

Hence you can infer each theory is partly but not completely true, which would also explain why each still has many adherents.

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