He adds his personal touch to the music.

has a musical touch to it e.g. signature tune

Regarding the ambivalent use of glätten here (s. my comment on a previous question) and the open criticism about that artist's 'accomplishments' uttered elsewhere by (presumably) the same author, 'Handschrift' takes on a rather ironic sense here: If the key virtue of an interpretation lies in the 'Glättung' e. g. of 'Ecken und Kanten' or the 'Schroffheit der (dynamischen, thematischen, figurativen etc.) Gegensätze' especially in Beethoven's sonatas, which his contemporaries often found offensive, the notion of "personal (style, fingerprint, whatever)" seems at best questionable. I think that the author meant to honor certain virtuousic achievements while reminding the reader that there used to be something more 'personal' present in the performance of earlier artists (such as, among pianists, Alfred Brendel).