I always saw it as a balance thing with Sonny. There is an eye for an eye policy in the Mafia, like in the last season of the Sopranos, Phil's brother's death only could be appeased by the death of Tony's cousin, Tony.

Although the Tattaglias were backing Solozzo, it's presumed, at least by this viewer, that Solozzo was fronting the deal and was the one responsible for the hit on Vito. It's implied that the Tattaglias backed the deal and Solozzo was the point man, hooking everything up cus he was the drug guy, so the negotiations and the hit were his doing on the surface. Although in actuality Tattaglia and Barzini were both behind the hit.

As per my Sopranos example, by killing Tony, Tony S. effectivly settled the score. By killing Solozzo the score likewise would have been settled. Although the Tattaglias backed Solozzo, the death of Bruno and Solozzo, especially Bruno, gave Tattaglia the right to go ahead and hit Sonny. This is elluded to in the boardroom scene when vito says "You lost a son, I lost a son". This is basically saying the score is even.

The hit on Bruno, caused by his hot headedness is what killed Sonny.