Is it possible to have a mecha act as a walking particle accelerator?
Yes, but it wouldn't make a very good weapon, for the most part.
The reasons for this are legion, and are covered on this site and others ad nauseam. The problem is that particle beams just don't make very good weapons because they are inefficient and short ranged due to a number of insurmountable issues that basically boil down to "nature abhors a plasma gun".
I came across this article on a new, tiny particle accelerator the size of a silicon chip.
It won't help you. The main problem with these kind of next-generation particle accelerators are their efficiencies which are low... much lower even than modern lasers. This isn't really a problem for research labs, but if you want to throw a death beam any distance and frazzle a tough target, it is a major obstacle.
Even if you did make a powerful enough plasma accelerator, you're still faced with the same problems as any other charged particle weapon.
If I needed to explain how the weapon works, would I point to this?
Doesn't seem necessary. Reactor provides power, gun eats power, produces pew pew.
Would the weapon work like laser ablation?
Probably not at all! What you have is a thing called attenuation, where an incoming particle incident on matter bounces off atoms in the target losing a little bit of its kinetic energy with each collision until it comes to a rest. This manifests as heating and disruption of molecular bonds and crystal structures some distance into the target, which should result in a cylinder or cone-shaped (depending on particle mass and velocity) region of melted or vapourised material. Surrounding material will be strongly irradiated.
Unless your particle beam is of stupendous power (and rest assured, it won't be, especially if you want to use it from something less than battleship sized and inside an atmosphere) the target won't go boom, alas.
A common weapon of these robots were lasers shot from the "eyes"
There's an additional problem here. Even used a fancy future super compact particle accelerator, you're still going to be hampered by trying to fit the whole assembly in the head. You could put it in the body and just put deflection and focussing magnets in the head instead, to turn the beam towards the enemy, but you're going to get additional bremmstrahlung losses which will manifest as additional radiation inside the head which is potentially bad news for equipment there.
What you should probably stick with are just boring old lasers. More efficient, easier and safer to use, easier to build turrets for. Laser ablation is exactly the sort of effect you should be looking for... Luke Campbell (who is an excellent source for all things laser death beam) refers to them as blasters due to the effect they would have on target material.