I've got to admit that for the longest time I tried and tried to come up with an interpretation of that verse that both 1) makes sense and 2) has anything at all to do with the context. I'm still working at it.
do you think the Pharisees could be the old wineskins?

(Here's the same passage in Mark and Luke)

The immediate context is John's disciples asking Jesus why His disciples don't fast. Jesus says the attendants don't fast while the bridgroom is with them but they will later when he is gone. No one sews unshrunk cloth as a patch on shrunk cloth. No one puts new wine in old wineskins.

It is that quick... no transitions or anything.

old wineskin: the law
new wineskin: grace through jesus' blood.

no one wants the new wineskin, the new covenant. they want the old. they like it. hard to change. but... God send His son and it changes things. Jesus is not a patch to be sewn onto the old traditions and ways. you need a new life in him. etc.

so i think. but anyway, i'm away from my bible and can't elaborate more for now about the context.