It could be that the system will spend such little time providing above 5 kW, once shading and losses are taken into account, that it's not worth the cost of getting the next-largest inverter.
It could be a restriction of your electricity network operator: often, they have bandings based on the maximum power that a PV system will deliver to the network, with different types of permitting applying to different bands.
Typically, there will be a cheap, easy permit for systems below a set small power rating, typically around 5 kW. And then a more expensive permit with a more time-consuming application process, for systems 5 kW - 100 kW (for example) . And one for 100 kW - 5 MW, and so on ...
The combination of these two things means that it's probably not worth your while having a system at 6 kW. A 5 kW inverter will be the most cost effective. And the extra 1 kW of panels means that you'll get best use out of your inverter - it will spend more time running at its peak output.
And no, you won't burn it out: the inverter won't draw more power from the panels than it can handle. It's completely standard to have a set of panels with higher nominal power than the rated inverter output.