Actually, both expressions are considered correct, but “on the one hand … on the other hand …” is preferred by many dictionaries while “on one hand … on the other hand” is listed as an alternative form.
However, intuitively, the first “the” seems illogical, because you are referring to one of your hands without specifying which one. You wouldn’t say, for example, “I wore a glove on the one hand and nothing on the other”, unless you were waving one of your hands while saying so, and you wanted to underline the fact that this had been the hand with a glove.
I’ve read an explanation that the first “the” originates really in gesticulation using one hand while saying the phrase (i.e. you would be looking at your hand and saying “on the one hand …”). Whether this is really true, we may never know.