Not sure what you're looking for. For all intents and purposes, this is probably just a simple graphic. If you can't get it to "look right", then you may simply need to keep trying. Like all animation, it may merely be a matter of getting the timing right Anyway, based on such vague descriptions it's impossible to make any sense of this. You need to provide more info on what you're actualyl doing, screenshots or even a clip on YouTube of the results you have so far.
When you try and duplicate an effect you need to carefully look at the shot, try and separate out the elements ad then start from there.
The first thing I notice is that there's some motion blur and that the graphic is more visible in the center of the frame than at the edges. The second thing that jumps out at me is that the graphic is barely visible and seems to blend in with the background colors, especially the lighter ones.
Start by handling the motion first. That's easy. Pick something easy to identify like the 1900 and count how many frames it takes to cross the frame. Set two keyframes for your graphic so that yours takes the same number of frames to move across the frame. Spin down the graph editor and check the speed value. How many pixels per second does it show? Now remove the second keyframe, go to the end of your shot and set another and adjust the X position until you get the same speed value for X. If your graphic isn't long enough or it's too short then fix the graphic. Enable motion blur and, if necessary, adjust the shutter angle in the Composition Settings to get the look you want.
Now let's handle the varying opacity. The easiest thing to use for this kind of feathered edge effect is a track matte. Create a new solid, put it above your graphic. use the mask tools to draw an oval mask that lines up with the graphic. Press the m key twice to reveal all mask properties and feather the mask. Now select your graphic layer and set the solid above as an Alpha track matte for the graphic. Adjust the shape of the mask and the feathering to get the look you want.
The last part is the easiest. The blending of the timeline graphic with the background makes me think that the blend mode for this layer was set to Screen. Press F4 until you can see the blend modes in the timeline and try that. If that doesn't give you exactly the look you like try others or try adjusting the overall opacity of the graphic.
There you go.
If you don't know how to do any of those steps please read the getting started materials or use the Search Help field at the top right corner of After Effects.