The engine house was built in 1872 to keep the 183 m (600' ) deep Towanroath Shaft free of water. Considering the Wheal Coates complex is situated on a remote cliff 61 m (200' ) above the sea, that's an impressive task. Other winding & stamps engines nearby lifted ore (copper and, primarily, tin) then crushed it for smelters elsewhere.
The mine closed in 1889, after a mere 17 years (the initial lease had been for 21 years), but a calciner building to the east of the complex indicates an abortive attempt to re-develop the mine between 1910 and 1914.

The landmark building, as seen in thousands of tourist photos, is now Grade II Listed and, like Wheal Coates' other two engine houses, has been stabilised and maintained by the National Trust. I understand there's an application to accord the remnants of the Cornish tin industry with World Heritage status, which should secure further protection and renovation.

Pedantic note: 'wheal' means 'mine', so 'Wheal Coates Mine' would be redundant; it's simply 'Wheal Coates'.
I presume the mine was named after its 'Clerk' (site manager?), a Mr. R.T. Coates.