The informal argument for this is as follows.
For a realisation x, determine its best chance of occurrence under H0 and also
its best chance overall. The ratio of these two chances can never exceed unity,
but, if small, would constitute evidence for rejection of the null hypothesis.
A likelihood ratio test for testing H0 : θ ∈ Θ0 against H1 : θ ∈ Θ1 is a test
with critical region of the form