gravity - How was the hypothetical ninth planet kicked so far out of the Solar System?

It's hard to say much about this planet, given that most of its properties are unknown. It hasn't been directly observed; instead, its effects on Trans-Neptunian Objects (TNOs) have been simulated and match observations. That said, its mass can be estimated, which is why it is conjectured to be the core of a giant planet.

One of the papers that Batygin & Brown cite is Morbidelli et al. (2012). Morbidelli has done prior work on the evolution of the Solar System if a 5th gas giant formed early in its history, which led to the Jumping Jupiter scenario. This is all described in a modification of the Nice Model.

Jumping Jupiter models with the four gas giants lead to one getting ejected from its stable orbit via interactions with the others (see
Nesvorný (2011)). This is one of the ice giants - Uranus or Neptune. However, if we postulate the existence of a 5th planet - a third ice giant - then it can be ejected from its orbit, leaving the other four in a stable arrangement.

Batygin & Brown are implying that the object perturbing the TNOs may be the core of this 5th gas giant.

Since writing the first draft of this answer, I've found out that Batygin has made some estimates that this planet would have been ejected long before the Late Heavy Bombardment - thought to be the result of a Jumping Jupiter scenario - occurred, meaning that it could not be that ejected ice giant, unless we accept a different cause of the LHB. This makes things more interesting, as different initial setups are possible.