A runny filling can leak from any cracks in the crust or wherever the crust may not be properly sealed. By running eggwash over the crust you can help to make a “glue” that holds the dough onto itself and creates a bond that can help to prevent some leakage.
You will also want to work quickly once the filling is added, to fold up the crust around the filling. If using frozen berries, the filling is more likely to leak as they thaw but working quickly and getting the fruit galette in the oven faster can help to set the crust so that it doesn’t become soggy.