What is verb tense consistency?

"To his great astonishment and mortification, Sticky saw his parents begin trying less and less to find him, instead devoting their time and energy toward the proper disposal of their newfound riches"

So I stumbled upon this sentence and the part which usually confuses is keeping the consistency of the tenses and my english teacher also points it out in my essays but tries in vain explaining it to me .

So the sentence sounds perfect to me, but still is there a reason against -" Sticky saw his parents began trying less and less to find"- being incorrect. Will be much obliged if someone can get me over with this.

Verb tense consistency is about using the same tenses to refer to one time period within a clause. In your sentence, this isn't an issue. The time is all in the past as determined by the verb 'saw'.

Here's a simple instance of non-consistent tenses where the middle verb is in the present while the first and last use the simple past tense:

*X bought a coffee, sits down, and opened her laptop.

To make the tenses consistent you could correct to

X bought a coffee, sat down and opened her laptop.

or

X buys a coffee sits down and opens her laptop.

But the verb forms in your sample sentence are fine.

Sticky saw his parents begin trying less and less to find him

'Begin' is not a tense marker here. The sentence follows a pattern where an infinitive (or -ing) verb form can follow the object of some verbs of perception: e.g.

I saw him fall.
I heard her turn(ing) the key in the lock.
She watched him leave.
Sticky saw his parents begin (...)