To support large-scale deployment of unmanned aerial vehicles (UAVs) in future, a new wireless communication paradigm, namely, cellular-connected UAV, has recently received an upsurge of interests in both academia and industry. Specifically, cellular base stations (BSs) and spectrum are reused to serve UAVs as new aerial user equipments (UEs) for meeting their communication requirements. However, compared to traditional terrestrial UEs, the high altitude of UAVs results in more frequent line-of-sight (LoS) channels with both their associated and non-associated BSs in a much wider area, which causes stronger aerial-ground interference to both UAVs and terrestrial UEs. As such, conventional techniques designed for mitigating the terrestrial interference become ineffective in coping with the new and more severe UAV-terrestrial interference. To tackle this challenge, we propose in this article new interference mitigation solutions for achieving spectral efficient operation of the cellular network with co-existing UAVs and terrestrial UEs. In particular, we exploit the powerful sensing capability of UAVs and inactive BSs in the network for interference mitigation/cancellation. Numerical results are presented to verify the efficacy of the proposed solutions and show their significant spectrum efficiency gains over terrestrial interference mitigation techniques.