Constructing a diverse repertoire of manipulation skills in a scalable fashion remains an unsolved challenge in robotics. One way to address this challenge is with unstructured human play, where humans operate freely in an environment to reach unspecified goals. Play is a simple and cheap method for collecting diverse user demonstrations with broad state and goal coverage over an environment. Due to this diverse coverage, existing approaches for learning from play are more robust to online policy deviations from the offline data distribution. However, these methods often struggle to learn under scene variation and on challenging manipulation primitives, due in part to improperly associating complex behaviors to the scene changes they induce. Our insight is that an object-centric view of play data can help link human behaviors and the resulting changes in the environment, and thus improve multi-task policy learning. In this work, we construct a latent space to model object affordances -- properties of an object that define its uses -- in the environment, and then learn a policy to achieve the desired affordances. By modeling and predicting the desired affordance across variable horizon tasks, our method, Predicting Latent Affordances Through Object-Centric Play (PLATO), outperforms existing methods on complex manipulation tasks in both 2D and 3D object manipulation simulation and real world environments for diverse types of interactions. Videos can be found on our website: https://tinyurl.com/4u23hwfv