This was taken near the tiger housing; the two layers of mesh seem a little closer together there, enabling the camera to be positioned a little closer to the (smaller aperture) inner mesh fence.
Then it was a case of choosing a long focal length and a wide(ish) aperture on the lens (although f/5.6 is as wide as I can go on that particular lens; and there is still a hint of the mesh pattern visible in parts of the out-of-focus background).
Also the subject needs to be as far away from the mesh as possible - in this case I was effectively shooting 'through' the open gateway which leads from the main enclosure, into the smaller outdoor area (adjacent to the housing) where Anastasia was laying, so I guess she was around three or four metres away.