SQL Server 2008 R2 - How does the SQL Trigger actually work in this case?
Trigger code executes in the scope of the transaction that invokes it. Since, as you say, each of these three records is inserted by a separate session, this means three different transactions were completed, each inserting one record and executing trigger code for that one record.

If you initiate three (or more) transactions "simultaneously", you cannot control the order in which parts of execute transaction will execute beyond setting the transaction isolation level. If the isolation level is set to serializable, then the trigger code will execute "one by one (according to the order they are saved in)"; otherwise it will seem to run "in any random order SQL seems fit", subject to whatever locks each transaction obtains.