Imagine a flat 2D being trying to understand a cube. You could show them its shadow (a square inside a square, connected at the corners), or push the cube through their flat world and let them watch the slice (a square that appears, holds steady, and vanishes). This page does exactly that, one dimension up.
Left: the tesseract's shadow — a 3D perspective projection. The small inner cube is the far cell, the large outer one the near cell; they're the same size in 4D.
Right: the slice — where the tesseract crosses our 3D world. Face-on it's a perfect cube: the cube that "holds steady" while the tesseract passes through.
Drag slice w slowly from −1.6 to 1.6 — watch the cube appear and vanish. Tilt the slice and it becomes a stranger shape, just as a tilted cube slices into hexagons. Alt-drag to turn the tesseract in 4D and watch the "inner" cube trade places with the outer. Click the slice to see which of the 8 cubes it came from.