Why use the word "down" for the sentence? Would it not have the same meaning when you exclude the word "down"? for example,

I lay close by her.

This sentence seems to have same meaning as, I lay down close by her. It's because a word lay means to put something or someone down.

Can (could) please explain this to me? Thank you.

'to lie' means 'TO BE in a horizontal position'.

Hence, when we move from the vertical (standing up) to the horizontal, we 'lie DOWN'.

"John lay down close to her" : he was upright and went down into a horizontal position next to her.

"John lay close to her." : John lay down, and so was then lying close to her. The sentence is stating what anyone would have seen AFTER he had lain down.

"If I tell anyone to "Hey, Lay to the ground", will have the same meaning as "Hey, Lay down to the ground""

No - you would say:
"Hey, LIE down ON the ground."

"Hey, Lay to the ground" has no meaning.
"Hey, lay IT on the ground" tells the person to gently place what they are holding/carrying on the ground.
Don't confuse 'to lie' ( where the Past Tense is 'lay')
and
'to lay'

"I still don't understand why the use of "DOWN" after word Lay, because it seems to be the same meaning regardless of having Down or not…"

No. Look carefully at what 'to lie' means: 'TO BE on a flat surface in a horizontal position'.
So the PERSON has to change from being vertical, to horizontal: to lie DOWN.

I think you are confusing 'to lie', with the meaning of 'to lay':

to lie :A person lies on the ground.
to lay: A person picks something up, or is holding something, and then PLACES IT DOWN gently.

A person lies in bed: A person(subject) lies(verb) in bed.
Note: THERE IS NO DIRECT OBJECT

If his mother brings him a morning cup of coffee:
She lays the cup on his bedside table = places it down gently on the bedside table.
She(subject) lays(verb) the cup(DIRECT OBJECT)…

I suggest you conjugate the two verbs so you can compare them, and then write some sentences for each tense, and post them so we can compare them.