Does anyone know something about the reasoning behind this? Has this something to do with cooling the jet exhaust "because stealth"?
No, I doubt it has nothing to do with cooling the exhaust. Typically the coolant, which in this case is essentially a type of synthetic oil, is used to cool high power electronics such as the radar system. Microwave radar is highly efficient at producing heat and also does a reasonable job of providing detection of other objects in the relative vicinity. The use of poly-alpha-olefins, introduced some 20-30 years ago, was because previous coolants were silicate ester based and would absorb water, break down and ultimately create a rather flammable alcohol to which bad things were prone to happen.
Interestingly this might partially explain a few things such as the need to "reboot" the radar by turning it off and on again. Perhaps it simply overheated because the coolant lines were too heavily insulated, undoubtedly due to non-conforming insulation which is the fault of a subcontractor, and didn't give off enough heat as they passed through the fuel. Shutting it off for a moment or two allowed it to cool enough so that it would restart normally. Of course that's just a bit of speculation on my part.
To address the comment of why the lines aren't on the outside of the aircraft, it's likely that at higher altitude there isn't sufficient air mass to dissipate the heat in a timely manner without using the entire skin as a heat exchanger. Of course that might raise issues with coolant loss when confronted with things like bullets.