Short answers, pulled from the story.
George Francis FitzGerald proposed a physical shortening of objects in 1889 to explain why the Michelson Morley experiment found no evidence of Earth's motion through the stationary aether. Hendrik Antoon Lorentz refined this idea in 1892 by suggesting that intermolecular forces behaved like electromagnetic ones to cause this contraction.
Albert Einstein published his theory of special relativity in 1905 and removed the ad hoc character from the contraction hypothesis entirely. He declared the concept of a stationary aether superfluous along with any absolutely stationary space and proposed that Lorentz's transformation apply to both electromagnetism and mechanics simultaneously.
Gamma equals one divided by the square root of one minus v squared over c squared where v is relative velocity and c is speed of light. At thirty million miles per hour which equals zero point zero four four seven times c, contracted length reaches ninety-nine point nine percent of rest length.
Magnetic force on moving charge next to current-carrying wire results from relativistic motion between electrons and protons. In electrons' frame of reference, moving wire contracts slightly causing protons of opposite wire to become locally denser and creating an apparent local imbalance that attracts extra protons.
Length contraction refers to measurements made at simultaneous times according to coordinate system unlike photographs taken from distance. Victor Weisskopf popularized result showing moving objects generally do not appear length contracted on photograph because a kind of visual rotation effect called Penrose-Terrell rotation occurs instead.