You must have misunderstood. Calls and texts are covered when you are roaming in the EU, i.e. when you are not in Germany, though Switzerland is not covered by this. If you call or text numbers abroad when you are in Germany then these are international and charged at the rate stated in the price list for your tariff. You will not be able to get these fees waived as you have been charged in accordance with the terms if the contract and the price list.

I need to make sure they aren't charging me for the month of January also, because I have not had service with o2 since the first of January, and I am using a different phone service with another company. The guy in the store said I could do that on the app, but I can't install the app because it's not compatible with my American phone.

What does ‘I have not had service’ mean in this case? Is there a problem with network reception at your location? If so, did you report it? What were you supposed to be able to do in the app?

Hello, it means they blocked my SIM card, so I have not been able to use my phone since January 1st. I am supposed to be able to cancel my service with o2 on the app. I can pay the fees for December, but I don't want to pay for January since I was without service the whole month.

Why?

If you have a bill in arrears of more than 100€, then o2 is entitled to block your tariff without the ongoing tariff fees. 
The information of the store employee can also be misleading. Perhaps o2 has canceled your contract and still demands the fees for the remaining term.  This corresponds to the legal possibilities in case of payment arrears. 
Therefore, you may have had to look for a new provider. 

Edit: According to the information from Andrea (see below), this has not yet occurred. Without a payment agreement with o2, however, the contract can be terminated without notice, but you still have to pay the contract costs (penalty).

Did you not receive any letters or mails or SMS from o2  regarding your bills?

Another explanation would be a block due to a conspicuously high bill for security reasons. This can only be unblocked by o2. 

You need to explain your situation in more detail as it is still not clear what your contractual position is.

Why did o2 block your SIM? If it was because you didn't pay your bill then you are still going to be charged because the contract is still running. You will need to pay the outstanding amount in order to have the SIM unblocked.

Termination of the contract is only possible by giving one month's notice to the end of the contract term. It is better to do this in writing, so you should send a letter to the address in Nuremberg that appears on the bill.

Dear @Leah15 ,

welcome to our community.

Your SIM card has been blocked because a large sum has been incurred that has not yet been paid. Of course, your contract will continue to run and must still be paid. However, the SIM card will only be activated again once you have paid your outstanding debts. Please contact our billing department via the hotline.

Thanks to @Klaus_VoIP and @bs0 for your help.

Kind regards,

Andrea