Morbius, a lot is involved when something like that happens!
First the "crackers" need to know your public IP address.Secondly, if they hacked into your computer it's probably that you are running some exploitable service, unsecure service( no or poor authentication, or whatever) or your computer's security just sucks! I'd say check your pc for spyware, viruses, etc.If you are running Windows and know little about security, learn how to do a basic pentesting on your home network. You are part of a community that teaches you how to do stuff like that, spend some time reading and reading some more.
If you get stuck, people here are cool, and with smart questions you will get smart answers.
Check your computer's logs for intrusions, learn how to use a firewall( i like Zonealarm, there's a free version out there). Keep in mind that security comes at a price!
Does your home IP address matches the one in their logs? Go to canyouseeme.org to find out what your IP address is and then tell them that you want to see the logs yourself, before you believe it, if they refuse, they are pulling your leg, or whatever...
This can be a long post, so I'll stop here.If you cover the above, you should have at least a basic knowledge of what's going on and know for sure what happend. If you find that someone hacked into your computer, save the log, do a print screen of the page, whatever it takes to save your a$$.
You can't have 2 identical public IP addresses on the Internet. Your IP would have needed to be "offline" for that person to use your public IP. I read something about this some time ago.
Also, your router's remote management, it's only to control the router remotely, not your computer. Ohh, and use a stronger password for everything.