What actually causes the bounce is quite a bit of physics calculations that haven't been completely solved yet (I hope to eventually make some more sense of it).
In simple terms though, it has to do with your kart's velocity in the direction towards the ground. As the kart advances you downward towards the ground it moves a certain distance each frame, eventually that frame ends up being at a point where your kart is actually below the ground and if it's a certain amount below, the kart will be ejected upwards giving you that slow bounce.
While this isn't "random," there's not exactly an easy way to avoid it. (Some old strategy guides recommend hold R before landing, this does nothing).
I'd recommend the following to most consistently prevent the bounce:
- Time Trials - Do nothing. Just drive straight off the ramp and you won't bounce.
- 150cc - Jump a little before the edge of the ramp. This needs to be at a pretty specific spot, but will reduce the number of bounces if done consistently. Here's a video of it in practice.
If you do bounce, be prepared to quickly triple tap the A button for fast acceleration to get back up to full speed a soon as possible.