Why is it that a vertically thrown ball will move horizontally if we are travelling in a non-inertial reference frame?

When you are on an accelerating train, you feel the acceleration because you are in direct contact with the train itself. As the train accelerates forward, the train's floor pushes on your shoes, making you accelerate along with the train. Both you and the train are rigid bodies in direct contact, with a high coefficient of friction between them, so any force applied to the train is transmitted directly to your body. It is the contact and friction between you and the train that allows you and the train to be treated as a single co-moving body.

A ball thrown in the air is not in contact with the train, so the train has virtually no means to impart an acceleration to the ball. This is basically the difference between pushing someone directly, and pushing the air a foot in front of them - you can't knock someone over by pushing air, as air simply does not transmit force the same way that a rigid body does. As a train accelerates forward, the air inside the train car doesn't accelerate the same way, the air will "slosh" to the back of the car since there's very little friction between the train car and the air. There is also very little friction between the air and the ball, so virtually no force applied to the train is applied to the ball.

It's very similar to putting a block of ice on a slippery plank - if you push the plank, the ice will mostly stay put and the plank will just move underneath the stationary ice. The low coefficient of friction only allows a very small amount of force to be transmitted before slipping. In the train and ball scenario, the train simply moves underneath the ball, as the air inside the train imparts very little force to accelerate the ball. If you put the ball in a seat, however, we're back to the rigid body scenario where the direct contact does allow the ball to be accelerated by the train.

Surely, the ball should feel the acceleration of the train just as much as the person on the train, in which case the horizontal location of the ball relative to the thrower should be the remain the same?

Yes, in the reference frame of the train both the ball and the thrower will be subject to the same fictitious force arising from the use of the non-inertial frame of reference.

The difference between the two is the presence of the real forces. The person experiences a friction force from the floor that is equal and opposite to the fictitious force. That prevents the person from accelerating in the non-inertial frame. The ball experiences only the fictitious force, so it accelerates as there is no real force opposing the acceleration.

If I throw a ball vertically inside a moving train, there will be horizontal movement if the train accelerates/decelerates (ie is not an IRF) and no horizontal movement if it does not (ie is an IRF).

Correct, in the reference frame of the thrower in the train.

Surely, the ball should feel the acceleration of the train just as much as the person on the train, in which case the horizontal location of the ball relative to the thrower should be the remain the same?

Once the ball is released in an accelerating train it no longer "feels" the acceleration of the train. The only real force acting on the ball is gravity and therefore the only true acceleration the ball will have is the vertical acceleration due to gravity.

However, in the reference frame of the thrower in the train, the ball will appear to accelerate horizontally as well as vertically. The thrower attributes the horizontal acceleration to some horizontal force acting on the ball (a pseudo , or inertial force). In actuality, assuming the thrower is constrained to move with the train by his/her seat or by static friction between his/her feet and the floor, it is the thrower that is accelerating horizontally and not the ball.

Hope this helps.