How can I achieve this effect?

Lens compression does not directly have anything to do with the focal length of a lens, though it may seem that way. In fact, it has everything to do with the distance between the camera, the subject and the background. You can achieve the effect by taking your distance from the nearby objects. Technically, cropping a photo taken with a 'normal' lens would have the same effect on the compression as using a telephoto lens.

How does it work?

First, about focal length. Focal length does not directly influence lens compression. It does have an effect on depth of field (bokeh) and the field of view. In fact, two photos taken from the same spot, one with a 35mm lens that was cropped and one with a 100mm lens, would show exactly the same size objects.

What does effect lens compression is the distance between the camera and the subject(s). In fact, it is the ratio of the distance between the camera and the subject(s) compared to the distance between individual parts of the subject(s) that decide how large these look relatively to one another. It is all about relativity!

Actually, taking your distance will make nearby objects smaller to a greater extend than far-away objects. To the human eye, it will seem that the nearby object grew smaller relatively to the far-away object. If you would think of relative sizes, you can say that the far-away object grew larger relatively to the nearby object. All you now need is a way to reduce the field of view, which can be using a lens with a longer focal length.

Further reading:

I suggest Elizabeth Gray's article about lens compression on Photographylife.com. She has taken some example photographs that make the effect of focal length quite clear.