The joke is, "How much does a polar bear weight? Enough to break the ice!" In english, "Breaking the ice" can mean making an awkward situation more comfortable. So, if you go up to someone you don't know and tell him that joke, you are, by simply starting a conversation, "breaking the ice."

It goes like this; You ask the girl/guy you like "How much does a polar bear weigh?" To which you most likely get an answer like "I don't know, how much?" You reply "Enough to break the ice." It serves as an icebreaker. "icebreaker" to relieve inhibitions or tension between people.(to start a conversation)

Unless I am missing the point - it is not a joke really - more of a pun, or play on words. It can be read two ways: It can mean "how heavy does a bear need to be before it will break the ice when it walks on a lake?" It can also sound like "How much beer do you need to drink, before you can become sociable?" That is because 'beer' sounds like 'bear', and 'break the ice' is an idiom for starting off a conversation.

'what does this joke mean' 'weigh' I don't think it is a joke - maybe a pun?