I heard the sentence 大好きなんですけれども. So my question is about the ending ～なんですけれども. What is the meaning of this ending, and when it should be used?
なんですけれども is widely used in polite conversation with an explanatory feel and maybe also argumentative texts. (My boss and also teachers used this a lot when they wanted to explain something.)
Its compounded of the explanatory expression んです/のです/の and けれども which is related to が/けど/けれど which roughly means but/on the other hand.
It is also used for weighing arguments, somewhat like "however".
大好きなんですけれども would mean "I really like it but...(on the other hand)" But could also as mentioned by user3856370 be used for polite refusal, or just a way of saying, "But i don't want to get into it right now" or "but there are reasons I..." As it often is with が at the end of a sentence, it is left to interpretation.
In the end the meaning and usage depends on the context.
んです is just the polite version of the 'explanatory の' concept (for which there are many entries on this site. And, けれども, simply means 'but'.
(The thing is), I really like you but...
You don't need anything after 'but'. It's a way of making an excuse. I'm guessing that this is a response to somebody's love confesssion, and it's a gentle way of letting the person down.