Eco-evolutionary dynamics are essential in shaping the biological response of communities to ongoing climate change. Here we develop a spatially explicit eco-evolutionary framework which features more detailed species interactions, integrating evolution and dispersal. We include species interactions within and between trophic levels, and additionally, we incorporate the feature that species' interspecific competition might change due to increasing temperatures and affect the impact of climate change on ecological communities. Our modeling framework captures previously reported ecological responses to climate change, and also reveals two key results. First, interactions between trophic levels as well as temperature-dependent competition within a trophic level mitigate the negative impact of climate change on biodiversity, emphasizing the importance of understanding biotic interactions in shaping climate change impact. Second, our trait-based perspective reveals a strong positive relationship between the within-community variation in preferred temperatures and the capacity to respond to climate change. Temperature-dependent competition consistently results both in higher trait variation and more responsive communities to altered climatic conditions. Our study demonstrates the importance of species interactions in an ecoevolutionary setting, further expanding our knowledge of the interplay between ecological and evolutionary processes.