The only way that I can make sense of this line is if Byron messed it up for the sake of the rhythm and the rhyme. If you imagine starting out with the phrase:
the first never knows the cause of the second
and then turning it around so that ‘cause’ is at the end of the line where it can rhyme with ‘applause’, then ‘the second’ gets replaced by ‘which’ and the result is:
of which the first ne’er knows the cause
but this lacks two syllables of a pentameter, and ‘first’ is hard to make sense of without ‘second’ to go with it. So you’d like to put ‘second’ back in somewhere, but there is nowhere good to put it. If you put it in the only place where it (just about) fits syntactically, you get:
of which second the first ne’er knows the cause
which is hard to understand and doesn’t scan. So I imagine that Byron said, “to hell with syntax!” and put ‘second’ before ‘cause’ where it makes sense semantically but not syntactically.
There is no doubt that the ‘second’ is the applause, but what is the ‘first’? This is ambiguous: it could be the poet (meaning that he doesn’t know that the audience are applauding out of drunkenness and good humour and not because they appreciate his speech), or it could be the audience, the ‘mess of men’ (meaning that they don’t know why they are applauding). The satire seems more biting if we take it to be the poet.