What happens when the transmission breaks while driving?

It will depend on whether it was an automatic or manual transmission.

Then it would depend on what broke.

If a clutch breaks you may see a lot of smoke and not be able to drive very fast.

If a pump breaks then you may coast to a stop.

If a linkage breaks then you may not be able to change gears OR you may have to select a different gear in order to drive at all.

If a gear breaks then you may hear some very expensive sounds and see some pieces flying around.