Impressive subject for a picture, @joseph. It was the right thing to do to get as close to the fence as you could to shoot through it. It blurs the fence so much that you can hardly tell its there.
Which underscores what @thefarmhand said about how to create natural blur in your pictures. The things he listed will help create natural blur in your image, and I’ll add a 5th one . . . increase the distance between the subject and the background, if you can. Obviously, in this picture, you couldn’t tell the snow leopard to change position like you could if you were photographing people, but there’s always the option to change your vantage point to increase the distance between the subject and background.
Run some experiments and watch the natural blur appear!