The mantle eventually solidified, causing the magnetic field to fail and let in radiation from the star it orbits.
Eventually is a very, very long time scale for this to happen naturally. A planet is more likely to suffer a catastrophic asteroid impact on these times scales (themselves very rare events) that would wipe out almost all life. Political developments (i.e. war) and improper environmental management (e.g. climate change denial) could wipe out a civilization far faster - hundreds of years perhaps. On the other hand the very slow loss of the magnetic field is something that could be adapted to and ultimately a technological civilization able to survive long enough to worry about this problem would possibly be able to fix the problem by generating some kind of EMF shield itself. You're looking at a K1 or K2 civilization on these time scales. Worst case scenario - you go underground and live in large atmospherically sealed environments. Or if you have the tech (K1 or K2) you could just forget the messy planet thing and live is large space stations. Planets are for young civilizations, not mature ones.
Incidentally we have few example planets to go on for life (one) and few examples of non-life bearing (as far as we know) planets, so it's by no means a foregone conclusion that a planet without a magnetic field or even an atmosphere will not support life. Life does not necessarily require a surface environment to evolve on or continue existing on. For example it is conjectured that life might exist below the ice of Europa in a liquid ocean.
How might the inner parts of a terrestrial planet cool and solidify without initially killings off life?
Exactly as they do now.
As AlexP points out the time between the magnetic field stopping and their being significant impact to life would be on the order of millions of years at least. So "initially" - no problem.
The loss of the Ozone layer does not prevent life from existing. For us it would be catastrophic (at the rate we're screwing up I think "will be" is closer to the truth) because we lack (and will do on the relevant timescale) the technology to adapt quickly enough on a mass scale to drastic climate changes - that's a different issue from yours.
This might be different if the planet had an extremely strong magnetic field (which brings it's own problems) and was close to a star with an extremely strong particle wind. But this combination presents a number of problems for making a life sustaining planet in the first place. It seems unlikely.
The core supplies almost no power to the surface compared to the Sun so the climate would be unaffected in any realistic scenario.