We say, "It's a matter of interpretation" when something is ambiguous or admits of multiple meanings (especially something requiring apologies and possibly lawyers). In the arts, "I loved your interpretation" commends the conductor, actor, dancer, writer or artist. It's easier to understand what artistic interpretation means when one can see it. Consider the following performances based on Camille Saint-Saëns' "Le Cygnet," ("The Swan"), a classic, especially in ballet repertoire, and notice the varied choices made in portraying the swan's movements.
The Russian ballerina Anna Pavlova (1881-1931) is perhaps the most famous interpreter of "The Swan." Appropriately, a meringue cake was named for her.
A more recent interpretation comes from another Russian ballerina, Maya Plisetskaya. Here she is in 1975...
...and again in 1986 at the age of 61:
It's possible that both ballerinas influenced L'il Buck, a practitioner of Memphis Jookin'.
"Swanlike" movements also inform the chinese State Circus's astonishing balancing act. (In this case the music is from Tchaikovsky's "Swan Lake.")
Interpretation includes parody, too. Michael Fokine dances as Ida Nevaseyneva:
Interpretation in an artistic sense can be understood simply as the fact that all artists make choices, and the sum of those choices constitutes an interpretation.