In Sonic Rivals 1, Silver learns that Sonic likes challenges.
Sonic meets Silver with playfulness, even though at this point, Silver's only interpersonal experiences come in the from of fights and conflicts. Silver brushes it off-- though he acknowledges as the game continues that Sonic is definitely skilled enough and powerful enough to take note of.
In Sonic Rivals 2, when they meet again, Silver remembers Sonic's playfulness and love of challenges, and uses it to his advantage. To make sure more chao are gathered where Eggman Nega can't get them, he challenges Sonic to a game to see who can get more Chao.
Sonic had been ready to meet Silver with the same seriousness Silver had met him with in the previous game, saying that he didn't have time for games. But Sonic readily goes back to playfulness as soon as Silver himself suggests making a game of the encounter.
From there, Silver sort of compartmentalizes Sonic as someone who likes having fun and is always looking for a chance to play games.
Which runs completely counter to Silver's whole life so far, with Crisis City and all. Silver's never been the type to think he had time for fun and games, or to even think about them at all.
But Silver gets a bit excited to show Sonic games that he might think are fun after that. He gets into it, and inadvertently starts having fun with it himself.
