Can I see Dota 2 game info online?
Solution 1:
Yes you can search dota 2 games by ID on dotabuff.com
Here is your game ... and it's pretty obvious why they won xD
However you won't see all the player names so you will have to know what hero he played,due the option in dota 2 to make your games publicly visible.
Solution 2:
You can use http://dotabuff.com/ (just search the match ID) to get a nice summary of the match.
The nickname won't always be displayed since each player can choose whether or not he wants to make its profile public (you'll need to know what hero your brother picked).