Kolmogorov complexity is, in essence, "How many bits do you need to specify an algorithm which will output the predictions of your hypothesis?" A hypothesis which gives a universally applicable formula is of lower complexity than one which specifies each prediction individually. More simple formulas are of lower complexity than more complex formulas. And so on and so forth.

The source of the high Kolmogorov complexity for the theistic hypothesis is God's intelligence. Any religious theory which involves the laws of physics arising from God has to specify the nature of that God as an algorithm which specifies God's actions in every situation with mathematical precision and without reference to any physical law which would (under this theory) later arise from God. As you can imagine, doing so would take very, very many bits to do successfully. This leads to very high complexity as a result.