While this is very stressful, you have not been charged and if what you say is accurate, then there is no evidence. If you live in an apartment or multi-unit building, perhaps there is some mistake. Either way, you might wish to contact an attorney and speak no more to anyone except through your attorney or after seeking his/her advice.
Additionally - It is very possible you might be the victim of identity theft. Check all your accounts. Change all your passwords. Change account numbers if possible. Change your email or at least change your email password.
I agree with Ms. Goldstein's response. Ordinarily I guess it would depend on who the "they" is that are asserting these claims against you. If it's an official (e.g., a police officer) I would take it more seriously. Even if you know you are not involved, it's always your best bet to consult with a seasoned criminal defense attorney before giving any statements. Identity theft and fraud can carry some pretty hefty penalties. While you likely have nothing to worry about if you have no involvement, it's best to have an advocate on your side.
One thought that pops into my mind is also whether there are others who live in your home that may have involvement? (Do NOT answer this question on this or any site; and I would advise against providing additional information to preserve your/their legal rights and defense if charges are brought). If so, they should contact an attorney right away. Credit card fraud can, and often does, arise from other parts of the country, and even other parts of the world.
As my colleague suggests, there is also the possibility you are an ID theft victim yourself. Please see this link for signs of identity theft and what you can do if you suspect you may be a victim: http://www.ohioconsumerhelp.com/sub/id-theft.jsp;jsessionid=A1E18BE00D127988CD891C6E98E7BE28.
All the best to you.
NOTE: This Answer does not constitute legal advice. Every case is fact specific. To render a legal opinion, an attorney must engage in a consultation with a prospective client and review any pertinent documents. This communication does not create an attorney-client relationship with Attorney Amy L. Wells or WELLS LAW OFFICE, INC.