Well, I can't say anything about the microscopic details, but in day to day language, the fast stream is adding extra material to the slow stream and it has to go somewhere. Since we are talking about flow that is confined it has to be up.
Let me try a goofy metaphor that should not be taken too seriously but does relate to everyday experience. You may imagine a couple of groups of people moving down a corridor, with a fast group overtaking a bunch of slow pokes. It may be that as they intersect people can just squeeze closer together an allow the young whippersnappers to go through, but if there is not enough room there will be a build up at the back. If instead of a corridor we imagine this interaction is confined on only one side, room can be made by having some people move farther from the wall than they started. That movement away from the wall is similar to a hydraulic jump.
Now let's look at the dynamics in a simpleminded way.
The fluid is deeper downstream than up, why doesn't it equalize? Whatever is happening at the jump itself, the fast stream is slowing down. That means that it is changing momentum which is to say it is subject to a force. The Newtonian reaction from that force is what keeps the increased depth from flowing back. Turn off the fast stream and the jump will flow back. The pressure available to support a jump will be proportional to the difference in velocities between the two streams.
It takes energy to raise the fluid, where does that come from? The fast stream is slowed down in the interaction, which means there is some kinetic energy running around loose. Some of that ends up raising some of the fluid.