What is the validity of the statement made by someone who says "Allahu Akbar, a word of truth" upon hearing the Adhan while he is speaking, claiming it to be a testimony to the truthfulness of his words?

There is no connection between the adhan coinciding with someone's speech and the truthfulness of that speech, nor is there any legal evidence for it. Similarly, the widespread belief among the general public that a speaker's sneeze is proof of their truthfulness is a false notion. The Sunnah for whoever hears the adhan is to remain silent and follow the muezzin by repeating what he says, then invoke blessings upon the Prophet (peace and blessings be upon him) and ask for al-Wasilah for him. Even if one is reciting the Quran, it is preferable to interrupt their recitation and follow the muezzin.