As explained in the other answer and in comments to the question, the responsibility lies with you. You had two separate contracts for travel from A to B and from B to C. You were delayed on A to B, but not enough to trigger any compensation, and you were not on time for check-in or boarding for B to C, so you were considered a no show.
In addition, you definitely had way too little margin, especially if you had checked luggage or for any other reason could not use online/mobile check-in. Unless you travel first of business class, the check-in/baggage drop deadline for LH at FRA is 60 minutes. This means you had one hour to:
- go through immigration (passport control) if you arrived from outside the Schengen Area
- wait for and reclaim your bags
- go through customs
- find and get to the check-in counter (which involves changing floors and possibly several hundred meters walk)
- wait in line
- check-in before the deadline
(after that you still have to go through exit passport control, security, and get to the gate before the boarding deadline, but if you made the check-in deadline it shouldn't be an issue).
Even if the plane was on time, not at a remote stand, you were among the first to deplane, you didn't have to go through passport control, and bags were delivered timely this would be quite tight, especially if you are not familiar with the airport.
If the plane was at a remote stand, you were among the last to deplane, you had to go through passport control, bags took a bit too long to arrive, or, as in you case, the incoming flight was late, then it's just not doable at all.
Note that this is very different from what would happen if you had bought both flights as a single ticket/booking. In that case, your luggage would have been checked through to the final destination, the only deadline is getting to the gate in time, so minimum connection times are much much shorter. And in that case, it would be the airline's responsibility to take care of you and get you to your final destination (and possibly compensate you).
So in your case, the airline has absolutely no obligation to help you. However:
You may ask the airline. Even though they have no obligation, they may want to help you by allowing you to rebook to another flight, either for free or for a fee (still better than having to buy a new ticket altogether). Don't count on assistance (hotel room, meals, etc.) or compensation, however.
If you bought the ticket through a site which specialises in selling separate flights for a single trip such as kiwi.com, they may have an insurance for you to cover precisely this case. Contact them for assistance.
If you have travel insurance (which you may have bought separately, or as an add-on when you booked, or may be part of the services of your credit card if you used one, or may be part of some other insurance or banking contract), check with them. The chances they will cover this are slim (as you had very little margin), but it's worth a try.
Also remember that if your FRA-NYC flight was not the last on your ticket (e.g. the outbound leg on a return ticket), then the airline may have cancelled the rest of the flights as well. Take that into account when considering your options.