In 3D you have up-down, left-right, and forward-backwards. You can move in any of those directions independently of the other two by pushing in the appropriate direction.
In 4D you have another axis available, call it “uft-rown”. You can push and move in that direction just like any other, maintaining the same position in the other three axes. It’s “simply” go the direction that *isn’t* up-down, left-right, or front-back, or any combination of those three. It’s just nearly impossible for us to visualize that because we think in 3d.
The cube on the wiki page is rotating, not translating (moving in a straight line).
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