How was coal key to industrialization?
It was the first fuel used to power factories.
Prior to the 19th century, most factories and mills were small operations because the relied entirely upon the force of water or air to power their machines.
The steam engine was first invented in 1606 by the Spanish but it was not until 1781 that James Watt invented a way to continuous spiral rotation, the turbine. From there the engine was adapted to be used for ever larger operations. Wood was the first fuel used but it is inherently uneconomical.
It was not until the 2nd half of the 19th century, however, that coal replaced wood as the primary source of fuel. It was also during the late 19th that America saw its greatest growth in industrial production.