I couldn't prove it formally but the main idea is that if you add P to all edges of some company, then the optimal MST (under the given constraints) itself doesn't change because the answer differs by a constant. If you add P to all edges of the first company for example, then the answer would be simply ans + P * K where ans is the answer of the unmodified problem.

This means that you have to find the optimal answer for some P. Obviously if you can find some P for which the MST has exactly K edges then you're done, but such P may not exist, so there is a trick explained in the last few sentences of the editorial explaining that you can simply find the smallest P for which the edges of company A chosen are less than K and then replace some edges with a formula.

I do understand why it works but I couldn't formally prove it. However, it is very intuitive once you understand the idea.