I specifically want to know why the past tense of the word 'have' is used in this phrase. In modern casual English(at least in the US), everyone says ' I should better get going' or they completely omit the word 'had' and say (and even write) 'I better get going'. I've read all posts on the internet regarding this phrase and not one single person questioned the word 'had' in this construction.
Cambridge says this but refrains from explaining the origin of the word 'had' in this phrase:
We use had better to refer to the present or the future, to talk about actions we think people should do or which are desirable in a specific situation. The verb form is always had, not have. We normally shorten it to ’d better in informal situations. It is followed by the infinitive without to:
Could someone please shed light on this?