How to infer that the black hole in M87 is rotating clockwise?

The ring of light is produced by plasma that is swirling around the black hole. Light (microwave emission at 1.3 mm wavelengths) is bent by the extreme gravitational field so that many light rays are bent in our direction by passing close to what is called the "photon sphere". i.e. The ring of light is not the accretion disk, it is a magnified halo of light caused by gravitational lensing of the general illumination surrounding the black hole.

However the black hole is likely to be spinning and its spin-axis is likely to be aligned with the large scale "jet" that is seen to emanate from the core of M87. The direction of that jet is more-or-less E-W as you look at the picture, but the jet is also coming towards us, with an angle of just 17 degrees between the jet axis and our line of sight.

The black hole is therefore likely to be spinning either almost clockwise or anti-clockwise as we view it from Earth.

The spin of the black hole means that plasma falling into it will be dragged in the direction of rotation at relativistic speeds. As it does so, any light emitted will be Doppler boosted in the direction that the material is moving.

That 17 degree tilt then becomes important, because there is a component of the velocity of the material circulating around the black hole that is either towards or away from us. Evidently, from the picture, we see brighter light to the south, which means that material circulating around the black hole is coming towards us from south of the black hole (and hence there is more light from here), but away from us to the north of the black hole. To arrange this, the spin-axis of the black hole must be aligned with the jet (i.e. it's projected spin-axis is roughly E-W on the picture) but pointing in to the picture, with most of the spin then being characterised as clockwise as we view it, but with a component rotating into the picture at the top and out of the picture at the bottom.