Given the prevalence of pre-trained contextualized representations in today's NLP, there have been many efforts to understand what information they contain, and why they seem to be universally successful. The most common approach to use these representations involves fine-tuning them for an end task. Yet, how fine-tuning changes the underlying embedding space is less studied. In this work, we study the English BERT family and use two probing techniques to analyze how fine-tuning changes the space. We hypothesize that fine-tuning affects classification performance by increasing the distances between examples associated with different labels. We confirm this hypothesis with carefully designed experiments on five different NLP tasks. Via these experiments, we also discover an exception to the prevailing wisdom that "fine-tuning always improves performance". Finally, by comparing the representations before and after fine-tuning, we discover that fine-tuning does not introduce arbitrary changes to representations; instead, it adjusts the representations to downstream tasks while largely preserving the original spatial structure of the data points.