Good question. I love this story, and I see it as being relevant and contemporary in several ways.
Fasting (or starving, in some translations) is the art the protagonist of the story performs. In the early days of his life, people are attracted to his work; they come and see him, they cheer for him, they are fascinated by him, and they are concerned that he eat once his fast was broken. Over time, though, he became more of a spectacle than an attraction. He eventually loses his position of prominence and is relegated to a walk-by attraction at the circus. People are more interested in the animals than a hunger artist. He eventually wastes away, barely distinguishable from the dirty pile of hay he's lying in.
Fasting is a metaphor, and it's open to some interpretation; however, we do know he sees what he does as an art, something he does because he must. As he's dying, he says,
"'I have to fast, I can't help it.... Because...I couldn't find the food I liked. If I had found it, believe me, I should have made no fuss and stuffed myself like or anyone else.'"
Fasting is a metaphor here for many things: something not everyone does, something which was once an art but is now out of favor with the public, the the thing one must do because it's just what one must do, something which requires sacrifice and commitment. That's still a relevant concept for any dedicated "artist" who practices his "art" despite (or in spite of) the approval or understanding of others. He feels isolated or "caged" by the dedicated, stringent pursuit of his art and the lack of understanding or interest by the masses.
The second point of relevance has to do with the uncaring, unthinking, fickle audience. Society is much more interested in that which is new (young) and fast and appealing and dangerous. The circus replaced the hunger artist with a panther. a "wild creature leaping around the cage that had so long been weary."
There was no art for this panther--they bring him food he liked, he didn't seem to miss his freedom, and the people were struck by his "joy of life." But we know one day the panther will no longer hold the attention of the circus-goers, and they'll move on to the next thing.
It is this juxtaposition between the passion of "art" and the fickleness of the masses which is still contemporary and relevant.