It is part of British 'understatement', that aside though

I know them quite well.

I know them well.

What is th difference in the meaning with the above sentences?

It depends, often in BE there is no difference. It is part of British “understatement”, that aside though “quite well” can also mean not to know the person as well as one could.

Could anyone rephrase the highlighted sentence so that I can understand that sentence?

Here is my suggestion:

It is an example of the British way of making things seem less important than they really are. Apart from that, however, ‘quite well’ can also mean…

Is it a bit clearer now?

Yes, Thank you very much.

I have got everything except (not to know the person as well as one could)

Could you explain this?

Sorry, that part seems so clear, it’s difficult to put it into even plainer English. I’ll try, anyway:

to know the person ‘less well’ than is possible.

or, put even more simply: not to know the person very well.

May be the writer was thinking of " I don’t know him quite that well" ? Maybe it’s a British thing, to me, “quite well” means “very well”, there’s no other meaning that I can think of.

I cannot really follow this thread, but I’ll chime in with the information that ‘quite well’ in BrE can mean either (1) ‘perfectly’ or (2) ‘reasonably well’.

From the online Oxford:

adverb 1 to the utmost or most absolute extent or degree; completely. 2 to a certain extent; moderately. 3 US very; really.

(Notice the more limited definition in AmE.)