length contraction
Length contraction is time dilation's partner: an object moving fast past you is measured to be shorter along its direction of motion than it is at rest. A spaceship 100 m long in its own hangar would be measured as, say, 50 m long as it streaks by at high speed. Nothing squeezes or crushes it; the ship feels perfectly normal to its own crew. It is the measured length in your frame that shrinks, and only in the direction of travel — its height and width are untouched.
Precisely: if an object has length L_0 when measured in the frame where it sits still (this rest-frame length is called the proper length), then in a frame where it moves at speed v its measured length along the motion is L = L_0 / gamma, where gamma = 1 / sqrt(1 - v^2/c^2) is the Lorentz factor. Because gamma is 1 or greater, dividing by it makes L smaller than L_0. At 87% of c, gamma = 2 and the object is measured at half its proper length. Like time dilation, it flows directly from the constancy of the speed of light and is completely negligible at everyday speeds.
The two effects fit together perfectly. Take the muon again: in our frame, the muon's clock runs slow (time dilation) so it survives the long trip down. In the muon's own frame, its clock is normal — but the atmosphere is rushing up at it, length-contracted to a fraction of its thickness, so there is far less distance to cross. Different stories, same outcome: the muon reaches the ground. Both frames must agree on what physically happens, even while they disagree about lengths and times.
A rod 1 m long at rest, flying past at v = 0.6c, has gamma = 1.25, so it is measured to be 1 / 1.25 = 0.80 m long in your frame — while remaining a full 1 m long in its own. Its diameter, perpendicular to the motion, is unchanged.
Only the length along the motion contracts; dimensions across the motion stay the same.
Contraction happens only along the direction of motion, never across it, and it is symmetric between inertial frames — each observer measures the other's ruler as the short one. It is a real measured effect, not an optical illusion, though what a camera would actually photograph is subtly different because of light travel-time.