The heavens are all but rent apart and the earth split asunder and the mountains brought crashing down. (Maryam 19:90)

تَكَادُ السَّمَاوَاتُ يَتَفَطَّرْنَ مِنْهُ وَتَنشَقُّ الْأَرْضُ وَتَخِرُّ الْجِبَالُ هَدّاً

The heavens are all but rent apart and the earth split asunder and the mountains brought crashing down.

(Maryam 19: 90)

 
Kaada is used to express that the verb it qualifies was on the brink of happening but didn't. To emphasize exactly how close it was to happening, imagine the verb being the man in the picture below, and the occurrence as the tightrope:






So in the verse you quoted, for example,


تَكَادُ السَّمَاوَاتُ يَتَفَطَّرْنَ مِنْهُ وَتَنشَقُّ الْأَرْضُ وَتَخِرُّ الْجِبَالُ هَدّاً
Whereby the heavens are almost torn, and the earth is split asunder, and the mountains fall in ruins,


The context is that of the Jews and the Christians claiming that Allah has a child.

So what is the link? al-Razi says there are a number of angles to it;

The first is that Allah is saying, "I would do this to the sky and the earth and the mountains out of My anger, were it not for My mercy!" (Imagine the scenario you often see in cartoons when a parent will say to their child when he has reached the end of his tether, "I'm so angry I could strangle you!" sometimes also accompanying the words by grabbing their childs neck. Of course they don't do it because it is their child, but they may apply pressure to the neck so that it hurts the child, and they are on the brink of doing it.)

The second angle is that this is a way of illustrating how serious and grave their claim is, and how is shakes and destroys the bases and foundations of tawheed [His Oneness].

And the third angle is that were the heavens, earth and sky to actually understand and grasp what was being said, then this would have actually happened to them because they would not be able to take the weight of the claim.

So the purpose of kaada in this verse, then, is to bring the even near to us and emphasise the seriousness of it in our understanding.