The only funny things we heard that evening came from the advertiser at the beginning of the programmer.
Why not use ‘The only funny thing’ instead of ‘The only funny things’?
The word "thing" is singular while the word "things" is plural. Thus, you would use the word "thing" when you are referring to one singular funny thing, and you would use the word "things" when you are referring to multiple funny things.
The word "only" does not imply that something is singular; rather, it implies a limitation. Thus, the phrase "the only funny things we heard" implies that while they may have heard many things in total, only a few of those things were funny. This particular sentence is expressing the idea that there was nothing funny besides the things that the advertiser said. Or expressed differently, every funny thing was said by the advertiser.
"Only" has a number of meanings, but in this context it defines some subset of a group. That subset can be any number. For example:
My sister's children are the only ones in my family who have red hair.
i.e., of all the children in my family, they are the only ones with red hair.
Those are the only houses in the neighborhood that don't decorate for Christmas.
i.e., of all the houses in the neighborhood, those are the only ones that don't decorate.
Your example is awkwardly phrased but semantically correct. The person is saying that, of all the things we heard in that evening's programme, the only funny things (plural) were from the advertiser.
I don't know the context but it seems to imply that the programme itself was supposed to be funny, but was not -- so much so that the advertiser was more funny than the actual content. Given the British spelling of "programme" this sounds like a typical example of British sarcasm.