is there any way to achieve such a smooth transition between two videos in the VSE, such as the transition that can be seen in the start of this video?
This particular effect looks a bit like the Transition > Wipe effect but in the Wipe effects it doesn't look like you can move the video, only reveal it with a mask.
You can put your 2 video strips on 2 separate tracks, select the second strip, open the N panel, and keyframe the Strip > Transform > Position X to make the image go from right to left:
If you want to add a kind of motion blur, you can select the strip and press ShiftA > Effect Strip > Gaussian Blur, which will add a strip over it, then select this effect strip and in the N panel > Strip > Effect Strip > Size X keyframe a high value to create some blur, then later in the timeline lower the value down to 0 and keyframe again:
If you choose to create an Effect Strip in order to have the blur effect, you can even create the movement with this strip (N panel > Strip > Transform), not with the original strip.