Integrating unmanned aerial vehicles (UAVs) into the cellular network as new aerial users is a promising solution to meet their ever-increasing communication demands in a plethora of applications. Due to the high UAV altitude, the channels between UAVs and the ground base stations (GBSs) are dominated by the strong line-of-sight (LoS) links, thus severe interference may be generated to/from the GBSs in the uplink/downlink, which renders the interference management with coexisting terrestrial and aerial users a more challenging problem to solve. In this paper, we study the uplink communication from a multi-antenna UAV to a set of GBSs in its signal coverage region. Among these GBSs, we denote available GBSs as the ones that do not serve any terrestrial users at the assigned resource block (RB) of the UAV, and occupied GBSs as the rest that are serving their respectively associated terrestrial users in the same RB. We propose a new cooperative interference cancellation strategy for the multi-beam UAV uplink communication, which aims to eliminate the co-channel interference at each of the occupied GBSs and in the meanwhile maximize the sum-rate to the available GBSs. Specifically, the multi-antenna UAV sends multiple data streams to selected available GBSs, which in turn forward their decoded data streams to their backhaul-connected occupied GBSs for interference cancellation. To draw useful insights, the maximum degrees-of-freedom (DoF) achievable by the multi-beam UAV communication for sum-rate maximization in the high signal-to-noise ratio (SNR) regime is first characterized, subject to the stringent constraint that all the occupied GBSs do not suffer from any interference in the UAV's uplink transmission. Then, based on the DoF-optimal design, the achievable sum-rate at finite SNR is maximized, subject to given maximum allowable interference power constraints at each occupied GBS.