I blocked someone on iOS 8.4, what exactly happens? More specifically, do our previous messages/conversations still show up on their phone? If not, is there anything I can do to somehow remove those messages from their phone?

Yes, anything you sent them before blocking them will still be on their device.

No, you cannot remove data from their device.

In addition, if you continue to send them messages, they will get them. Blocking is one way. You can call them and message them. If they call you, they will be sent directly to voice mail and the voice mail will be filed away under "Blocked Messages". You will not be notified of it. SMS, MMS, and iMessages from them will be discarded. You will not receive them.