It is not obvious from the illustration, but the crankshaft and the shaft carrying the cylinder are offset from each other. Therefore as the flywheel and the cylinder rotate the piston in the cylinder moves in and out. The only real advantage seems to be the compactness of the engine.
Once again it is not clear in the picture, but the cylinder is only supported on the side nearest the viewer. There must have been large sideways loads on the bearing.
Many thanks to my correspondent Clive for bringing this to my attention.
Source: Practical Magazine p432 of Vol 2, issue 12, 1873. The original publication seems to have been in the Scientific American for April 5th 1873. The picture and text is the same.