How did the wise men (Magi) know the country where the new king was born? After Jesus was born in Bethlehem in Judea, during the time of King
  Herod, Magi from the east came to Jerusalem and asked, “Where is the
  one who has been born king of the Jews? We saw his star when it
  rose and have come to worship him.” (Matthew 2:1-2, NIV)


I can't understand how one can conclude that a single star is pointing at a specific location. Even if I draw some lines by joining some stars, I don't think I can point at a specific location on earth. Even if I think that it points to a particular place, when I reached that place, the same line will point again somewhere ahead from that location.

The wise men were from the east, indicating that they were not Jews, I presume. They simply saw an unusual star and are able to calculate or know that a new king of the Jews is born in Israel. From the appearance of the star, I believe it is possible to conclude that a new king is born, provided the observer truly believe in star signs. However, I still can't understand how the star provided some clue where the new born king is. Moreover, the wise men were very specific when they asked king Herod. They said "Where is the one who has been born king of the Jews?".

Why did the wise men stop at Jerusalem and asked "Where is the king of the Jews"? They could have gone up to Egypt or Greece, or even further. From their question, it sounds like they were searching in Israel only. It appears as though they knew that the king was born in Israel and hence they went straight to Jerusalem, the city of the king of Israel.

How did the Magi know the country where the new king was born by simply looking at a distant star in the sky?
I think what you're looking for is detailed in Matthew 2:1-12:


  After Jesus was born in Bethlehem in Judea, during the time of King Herod, Magi from the east came to Jerusalem and asked, ‘Where is the one who has been born king of the Jews? We saw his star when it rose and have come to worship him.’
  
  When King Herod heard this he was disturbed, and all Jerusalem with him. When he had called together all the people’s chief priests and teachers of the law, he asked them where the Messiah was to be born. ‘In Bethlehem in Judea,’ they replied, ‘for this is what the prophet has written:
  
  ‘“But you, Bethlehem, in the land of Judah,
      are by no means least among the rulers of Judah;
  for out of you will come a ruler
      who will shepherd my people Israel.”’
  
  Then Herod called the Magi secretly and found out from them the exact time the star had appeared. He sent them to Bethlehem and said, ‘Go and search carefully for the child. As soon as you find him, report to me, so that I too may go and worship him.’
  
  After they had heard the king, they went on their way, and the star they had seen when it rose went ahead of them until it stopped over the place where the child was. When they saw the star, they were overjoyed. On coming to the house, they saw the child with his mother Mary, and they bowed down and worshipped him. Then they opened their treasures and presented him with gifts of gold, frankincense and myrrh. And having been warned in a dream not to go back to Herod, they returned to their country by another route.


So, they headed first West (because of the sign of the star), then to the capital of the country they found themselves (Jerusalem), then zoomed in further by heading for Bethlehem (on the advice of Herod's staff) and finally to the exact house, because the star was above it.

(Perhaps they should simply have followed the star all the way, instead of detouring via Jerusalem?)