Piano strings are stiff, and stiff strings do not behave like a perfect ideal string. Their overtones run slightly sharp, more so at the top of the keyboard. If you tuned every note to exact math, the piano would actually sound out of tune to the ear. Stretching the tuning at both ends corrects for this.
A well-stretched tuning sounds fuller and more alive. Octaves lock together, the bass has depth, and the top notes sing rather than sounding thin. On older and player pianos, careful stretch tuning brings out character that a flat, by-the-numbers tuning misses.
Stretch tuning is guided by instruments but finished by ear. It takes judgment built over years of tuning many different pianos. Clare's has done this work for three generations.