You need to set the inbound server type as IMAP instead of POP3.
With IMAP the master copy is held on Hotmail's servers and your PC is synchronised with that when you are online.
With POP3 your PC is the master copy and any new emails are downloaded to your Inbox. It's then up to you where (i.e. which folders) you put them on your PC.
It's nothing to do with (i.e. not unique to) Thunderbird, - it's how most email clients work.