If the transcription is correct (ignoring punctuation/capitalisation, which has gone a little awry) then it most probably refers to a clicking sound of one's heels on the stairs, probably suggesting a brisk pace. However, "I have clicked up them" doesn't seem to fit, "stopped to " should presumably be "stopped at ", and I don't see how the speaker can arrive at the bottom of the stairs having climbed up them. These things make me doubt the accuracy of the transcription.
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gambolerWell, I came out of the door and walked across the hall to the stairs. Then, I heard click up there. I stopped to the bottom of the stairs and called out: "is anyone there"?It would have to be "I heard a click up there".
gambolerAlthough I agree with you with "at" sounding more natural than "to", I googled the expression