Why was Jesus able to silence his critics simply by pointing out that the Messiah was both Lord *and* son? In Matthew 22, after basically frustrating the Pharisees and the Saduccees by answering some really tricky questions, Jesus finally turns the tables on them and asks this question: 


  41 While the Pharisees were gathered together, Jesus asked them, 42 “What do you think about the Messiah? Whose son is he?”
     “The son of David,” they replied.
   43 He said to them, “How is it then that David, speaking by the Spirit, calls him ‘Lord’? For he says,
     44 “‘The Lord said to my Lord:
     “Sit at my right hand
  until I put your enemies
     under your feet.”’[e] 
   45 If then David calls him ‘Lord,’ how can he be his son?” 


The response is silence, and apparent victory:


  46 No one could say a word in reply, and from that day on no one dared to ask him any more questions. 


What I understand is this - Jesus is pointing out that the Messiah is both David's son and David's Lord.  I get that its a good theological point.  But why does it silence his critics?
Earlier on in Mathew 22 the Pharisees are trying to trap Jesus with what they probably considered paradoxes or no-win scenarios (ie Jesus picks one side he'll anger the other side). I've always imagined these questions are the kinds of questions the Pharisees argued over in the internet forums of their day (Should we pay taxes to Caesar? Matt 22:15, Will the Dead will rise again? Matt 22:23, What's the most important commandment? Matt 22:34).

The question Jesus asks the Pharisees in verse 41 is along the same lines. He shows them something that was probably ingrained in them since they were born (that the Messiah is the Son of David) and called it into question. In a way it challenged their thoughts on God and who the Messiah was really going to be. And exactly like you mentioned in your comments the theological implications were not lost of the Pharisees, who as far as I know, never expected the Messiah to be both God and Man. Imagine your own theology being blown out of the water by your own scriptures to the extent of being irrefutable, by virtue of source and being logically sound. This wasn't something Jesus was just saying (something that could be refuted), he instead pointed them to scripture which is something they all held sacred. 

Hence the silence.