Exchange Online: how to use Partner to Office365 connector

Question

I try to create a service, that gets emails from tenants, edits them by some terms, and places them in the relevant email accounts on the tenant exchange online.

I found that I should do it with a connector. I set the connector for the "inbound" traffic, with my service DNS and everything, but when I try to send an email to "smtp-mail.outlook.com" at port 587, I get this error message:

530, 5.7.57 Client not authenticated to send mail

Does anyone know how should I use this type of connector? I can't find any docs about how to send a message using an inbound connector.

I wish to send those emails, when the from and to addresses stay the same, and not like they were sent from some kind of service.

Thanks!