It is very easy to do with selective focus. Shooting with a wide aperture and creating a shallow depth of field is an effective way to feature your subject and make the background into a soft, creamy blur. Positioning your subject a greater distance from the background and using a long focal length is the other common option. Most everyone knows these suggestions.
Framing the subject off-center and creating negative space works wonderfully too. I think too many underestimate the importance of the background. I look at the background first in many cases. How light washes the scene. Do you think the background a crucial element in your composition?
A valuable tip that many forget is to get as low to your subject as possible. Just this slight adjustment of your position to the subject will eliminate the busy vegetation and other ground distractions. Getting down to eye level with the subject often creates a far more compelling image too. The inconvenience of dropping into mud, dirt and snow, plus the difficulty should not deter you. The angle of view is likely worth it.
I think getting eye level with the subject is likely the best tip. Below, I am just slightly above this swallowtail. The technical tips of blurring the background are just so common, where anyone taking photos is likely to know them, but not everyone thinks about bending those knees.
Below, the group has different focal lengths and different lenses. It is all in how you see your subject in what you choose to include in the final image.
The three images below all have a different way of seeing the subject. The snail photo shows where the snail is resting, so the lily is of equal importance – and has similar coloring too.
The grasshopper is more about the story of it eating the flower. I think each background supports the subject.