Other forms: building bridges; built bridges; builds bridges
To build bridges is to improve relationships between people or groups who are different, disagree, or have been in conflict. It's better to build bridges with people you disagree with than to devolve into all-out conflict.
A literal bridge may connect people on one side of a river to people on the other side, allowing interaction, trade, and a better understanding of each other. That's the idea behind the figurative use of building bridges: improving communication and positive interaction between often opposing, or at least very different, sides. A company might organize a fun retreat to help build bridges between different groups in the organization, such as engineering and marketing, hoping to foster team cohesion. The opposite of building bridges is burning bridges, cutting off connections.