What are the draws you put on in winter?
I genuinely have no idea what you are referring to there, presumably a regional thing.
An explanation.
"Winter draws on" is a phrase that means that winter is coming. There may also be the implication that it is necessary to take precautions.
"Winter drawers on" is a command. It means "Put on your winter-specific underwear". This may be a precaution that one takes to avoid getting chilled in winter.
Thus we have two different interpretations of the same phrase, one of which alludes to underwear (drawers) and may cause amusement. This is called a "pun". One definition of a pun is a "joke based on the interplay of homophones". In this case we have the homophones draws (approaches) and drawers (undergarments).*
The result of this is hilarity.
OK. Well apart from the fact those are still "drawers", not "draws",
Yes - that's the point of the joke. Two words that have different meanings (and may even be spelled differently) but sound the same.
I hope that this explanation has been useful, and will enable you all to laugh politely next time you hear this, rather than looking baffled and saying "But that doesn't make sense. It's not spelled like that. I don't understand."
“Explaining humor is a lot like dissecting a frog, you learn a lot in the process, but in the end you kill it.”
― Mark Twain
* I wouldn't be at all surprised if there are regional pronunciations in which these words are not homophones. I know that the Scots tend to pronounce some letters in words where that pronunciation has been lost in English regional pronunciation. For example the W in sword, and the R in moor. So it's possible that this fantastically funny joke may fall a little flat if told in Scotland.