The increase in (relativistic) mass can be explained by light speed constancy. The faster you travel, the more distance you actually need to cover to travel through the same amount of space-time. Therefore, to accelerate, you need to put in more energy than you initially did to obtain the same acceleration at slower speeds. Therefore, for f=ma to continue working at high speeds, the "m" needs to be increased. Mass can be viewed as resistance to change in momenutm, and as you can see, the resistance to change in momentum will be greater if you need to put in more force to accelerate it.

The particle, however, does not feel its mass increasing, thus why it is called relativistic mass.

(as the particle could say that it is the one at rest and the observers are travelling towards it at light speed).

If I rephrase your question slightly- if the train was travelling at 99.9% the speed of light (as because it has mass, it is not possible of it to travel at the speed of light) and you ran through it at 10km/h, the speed of you relative to the observers will becaome something like 99.9000000000000000000...etc..001% the speed of light (or something like that). This is due to the warping of space time (time would distort so that you would not break the light speed barrier- as to both you and the observers, the speed of light is constant).

I think i was wrong to say that you would not be able to run at 10km/h if you were travelling very fast. If you consider the train to be stationary and the observers to be travelling towards you at 99.9% the speed of light (as you are perfectly entitled to do due to relativity), then you could obsiously easily run at 10km/h, so this must still be true even thought the train is travelling at near light speed. However, your relative velocity to the observers would not change by 10km/h due to the distortion of space and time.

"I have great faith in fools; self-confidence my friends call it."

-Edgar Allan Poe