The examples using the ConfigurationManager are certainly relevant when trying to read configuration information for the current running web site/application. However, there are cases when an application needs to manage the connection strings from a web.config file from another location.
Configuration conf;
ExeConfigurationFileMap fmap = new ExeConfigurationFileMap();
fmap.ExeConfigFilename = @"C:\AdifferentProject\web.config";
conf = ConfigurationManager.OpenMappedExeConfiguration(fmap, ConfigurationUserLevel.None);
ConnectionStringSettings MyConnSetting = conf.ConnectionStrings.ConnectionStrings["MyDbConnString"];
"MyDbConnString" is the name of a connection string in web.config