To understand the operation of a blazed grating you must first be familiar with the basic mode of operation of a refraction grating.
Consider a beam of monochromatic light incident on the grating. The rulings of the grating are illuminated by the incident radiation and scatter or reflect the radiation, acting as independent but coherent sources (i.e. the sources have a fixed phase relationship to each other). The radiation from these sources then interferes in the region in front of the grating producing regions of maximum and minimum intensity. Maxima and minima occur along lines at definite angles relative to the plane of the grating, and the maxima can be considered to be output beams of different orders.
Gratings can be either transmission types or reflecting types. In the former the illumination passes through the body of the grating before encountering the rulings, and the output beams are produced on the far side of the grating opposite to the incident side. In a reflection grating the surface of the grating is coated with a reflecting coating - usually thin film of aluminium, and the output beams are produced on the same side of grating as the illuminating beam.
In both types, the locations of the orders (i.e. the angles of the output beams) are determined by the pitch of the grating, the incident angle of the illuminating radiation, and its wavelength. Whether the rulings are blazed, or, if they are, the blazing angle, has no effect on this. This is an important point and must be remembered when reading what follows.
In an unblazed grating the geometric form of the rulings (apart from their spacing, that is) is not controlled so as to conform to any particular shape, so they reflect or scatter incident radiation in an uncontrolled way, perhaps more-or-less equally in all directions. But in a blazed grating (which is normally a reflecting type), the rulings are shaped so that, with a given angle of incidence, light is preferentially reflected in a particular direction according to the usual rule for specular reflection. This direction (which depends on the incident angle, but not on the wavelength) is chosen to be equal to the output beam angle for a preferred order and wavelength, as produced by interference as described above. In this way the rules for diffraction and for reflection produce the same result - an output beam at the same angle. The blazing of the rulings anticipates, as it were, the beam angle produced by diffraction and interference, and concentrates light into the direction of the output beam. This direction is actually controlled by the spacing of the rulings according to Bragg's law, but the blazing angle is chosen so as to produce the same result for a particular wavelength and order.
Note that blazing can only optimise a single order and wavelength, since the angle of the diffracted beam varies with wavelength and order, whereas the angle of the (putative) reflected beam is fixed by the incidence angle and is the same for all wavelengths.