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Channel knowledge map (CKM) has recently emerged to facilitate the placement and trajectory optimization for unmanned aerial vehicle (UAV) communications. This paper investigates a CKM-assisted multi-UAV wireless network, by focusing on the construction and utilization of CKMs for multi-UAV placement optimization. First, we consider the CKM construction problem when data measurements for only a limited number of points are available. Towards this end, we exploit a data-driven interpolation technique to construct CKMs to characterize the signal propagation environments. Next, we study the multi-UAV placement optimization problem by utilizing the constructed CKMs, in which the multiple UAVs aim to optimize their placement locations to maximize the weighted sum rate with their respectively associated ground base stations (GBSs). However, the rate function based on the CKMs is generally non-differentiable. To tackle this issue, we propose a novel iterative algorithm based on derivative-free optimization, in which a series of quadratic functions are iteratively constructed to approximate the objective function under a set of interpolation conditions, and accordingly, the UAVs' placement locations are updated by maximizing the approximate function subject to a trust region constraint. Finally, numerical results are presented to validate the proposed design achieves near-optimal performance, but with much lower implementation complexity.