You get the error because you delcared the property System.Data.Common.DbConnectionStringBuilder. This would say that is returns and takes a DbConnectionStringBuilder object. Yet you try to convert it to a String object, which it is not.
I suppose I don't understand the use of the class itself. It seems like a wrapper that isn't needed. You could just as easily use a Dictionary<string, DbConnection>. This way you could do like so:
Seems a little easier. Also, is there any perticular reason for the user of DbConnectionStringBuidler?
var dbConnections = new Dictionary<string, DbConnection>();
dbConnections.Add("LocalDb", new SqlConnection("myconnectionstring");
DbConnection localConnection = dbConnections["LocalDb"];
var cmd = new SqlCommand("Select * FROM Mytable", dbConnections["LocalDb"]);