3. aA strip of wood, iron or other material fastened across something to give strength, hold in position, furnish a grip, etc.; as, a porcelain cleat with grooves in which electric wires are fastened; a strip of leather fastened to the sole of a shoe to give a firm grip. b Specif.,Joinery, a frame of wood or iron used instead of cramps for compressing joints, etc.; also, a chock or bearing block.
MyWebsters New Dictionary, 2d ed, 1952, gives at least three meanings for "cleat", only one of which (posted above) relates to cleat in woodworking. For an "insider" to woodworking, visualizing a cleat, and how it functions, is easy; describing it for "outsiders", though, is not so easy.
... The seat may be made in two ways; cross pieces of wood may be used or a softer seat with webbing and wire springs. If the former is chosen the cross pieces should be 24" long, 3 1/2" wide and 5/8" thick. Mortises of full size to receive the ends are cut on the inside of the side pieces 2" from the lower edge and placed as shown in Fig. 1. If springs are used four strips of 3" webbing are run both across and front and back, the ends being securely held by cleats 1" square which are screwed to the inside lower edge of the cross pieces. Short spiral wire springs are then sewed where the webbing crosses, 16 being required. The tops of the springs are then secured by adŽditional strips of webbing or by a piece of strong canvas, the latter preferred. Allowance must be made for the depression of the springs and canvas caused by the weight of the person occupying the chair.
Source: JOHN F. ADAMS.,"A Reclining Chair", Amateur Work May 1902 page?