In posing questions such as this it is helpful to mention whose ruling we are interested in. If the interest is in the general treatment of the issue in Shi'a Fiqh, the generality usually in the area of Fiqh and Naqhl (narration) brings ambiguity. Having said this I believe ruling regarding chess is different depending on whose ruling we are talking about.
As it has already been mentioned according to Imam Khomeini playing chess as a sport or hobby is not Haram. Since the questioner is more interested in knowing the reasons for prohibition, I think a correspondence between Imam Khomeini and one of his student, Ayt.Ghadiry might be of some help but a more elaborated discussion of the reasons must be sought in more specialized sources and from scholars who are more qualified in the field of Fiqh.
We may want to know that someone wrote to Imam Khomeini and said that chess was considered a sport and it was a mental exercise and he wanted to know if playing chess was Jayz according to Imam. Imam Khomeini wrote that it was Jayz and the reply was widely broadcasted in Iranian Mass Media then. Ayt. Ghadiry in a letter to Imam Khomeini wrote that he thought Imam was wrong in permitting chess because chess is widely played for gambling purposes and had no significant sport function. This letter was also publicized widely by those who found the ruling wrong. In yet another publicized letter Imam Khomeini wrote a very short answer to Ayt. Ghadiry that he was surprised to read his letter and that expected more competency in Fiqh from him. He told Ayt. Ghadiry that by then he should have known that ruling in Figh are context bound and answers are only meaningful in the context of the questions being asked. And if the original assumptions of the question were different the ruling would be different (please note that this a free translation from my memory.)
As we see the basis for this ruling was whether chess is a means of Haram (gambling) or not. There is a Mabna (theoretical foundation) in Fiqh which some scholars accept and some do not it says that any means of a Haram activity is Haram itself, those who do not believe in this theorem do not consider playing chess Haram if the purpose of playing is not gambling,
In his letter to Ayt. Ghadiry Imam Khomeini mentioned to him that Ayt. Khonsari, who was one the most mottaghi (righteous) person he has seen, did not consider playing chess Haram. Apparently Ayt. Araki did not believe in the above theorem since as I remember he did not consider playing cards Haram given that gambling was not involved in the play, while we know cards are widely played for gambling purposes. However it is clear from Imam Khomeini's correspondence with Ayt. Ghadiry that had chess been predominantly a gambling instrument he would have considered it as Haram even if in individual cases it was not used for gambling purposes.