How does the filter work? Does it simply catches all emails, scan and inspect them, then pass to our mail server?

Yes exactly, all incoming email traffic is received by MailChannels, filtered and then passed to your mail server. You need to set 2 things for the inbound filter to work: point you mx records to us (mx1.mailchannels.com and mx2.mailchannels.com), then, create an Downstream record that points to your servers from inside of your domain console settings area.

When mail from your domain gets sent to us (because you pointed your MX record to us), we then look up where it should be delivered by looking up the entries you provided for Downstream Records.

Then you can set up spam handling policies, such as rejecting the message, quarantining the message, or flagging the message. Senders are notified, in the event the message is rejected or quarantined. Recipients are notified, in the event a message is quarantined, via a periodic quarantine digest email. The quarantine digest contains a summary of messages that were quarantined since the last digest was issued, and contains links to release the message from the quarantine, block the sender and whitelist the sender. The recipient will not be notified if the message was rejected or flagged.