Developers write source code in human readable programming languages which is then needs to be processed further so it can be deployed, for example, to a production server.
These steps are usually called build steps (or "the build").
Many teams use tools to automate these steps. So that they will be reproducible and easy to execute.
Some teams run these tools manually after changes to the source code are made but others use continuous build systems where a server automatically detects that a source change happened and it runs those build steps.
When people say "don't break the build" they usually mean - don't make changes to the source code that will cause the build steps to fail.
When that happens it blocks all the developers from building new versions of the code until the offending code gets fixed. Depending on the team size this could be very costly.