They knew that eventually, the whole thing would be played. You can't offer part of evidence without the other side being able to offer the rest if the evidence that provides context. But instead of playing a snip and then forcing the prosecution to play the whole thing during rebuttal, which would make it look like the defense was hiding something and playing games, the defense played the whole thing.
My feeling is either Jodi insisted on this evidence being offered against advice of counsel or the defense is just grasping at straws because their case is so very thin.
I thought they were convincing people until they introduced the tape, though. One other possibility: perhaps the prosecution intended to introducd it in rebuttal to prove what we heard- that Jodi was a willing participant. So they felt they had to present it first to frame it differently. If so, it didn't work.