An oath is a formal declaration or promise made by a person to affirm the truth or fidelity of their statement or the sincerity of their intentions. Saying "I" signifies the personal accountability and commitment of the individual taking the oath. It emphasizes their willingness to be held responsible for the truthfulness of the information they are about to provide.
There is nothing special about the letter "I" itself; it simply serves as a grammatical indicator that the person taking the oath is speaking on their behalf and personally affirming the truthfulness or faithfulness of their statements.