You can think of an engine being switched on. Somebody turns the key, the engine coughs, and the thing begins to run. Everything about this word lives in that first minute. A start-up is the company that has just been switched on: it is moving, but it has not gone anywhere yet. And start-up costs are what you had to pay before the key would even turn — the fuel, the parts, the licence. Both uses are about the same short, expensive, hopeful moment at the beginning.