Transfer of objects between humans and robots is a critical capability for collaborative robots. Although there has been a recent surge of interest in human-robot handovers, most prior research focus on robot-to-human handovers. Further, work on the equally critical human-to-robot handovers often assumes humans can place the object in the robot's gripper. In this paper, we propose an approach for human-to-robot handovers in which the robot meets the human halfway, by classifying the human's grasp of the object and quickly planning a trajectory accordingly to take the object from the human's hand according to their intent. To do this, we collect a human grasp dataset which covers typical ways of holding objects with various hand shapes and poses, and learn a deep model on this dataset to classify the hand grasps into one of these categories. We present a planning and execution approach that takes the object from the human hand according to the detected grasp and hand position, and replans as necessary when the handover is interrupted. Through a systematic evaluation, we demonstrate that our system results in more fluent handovers versus two baselines. We also present findings from a user study (N = 9) demonstrating the effectiveness and usability of our approach with naive users in different scenarios.