How to explain the strange behaviour about Function Nothing

Taking a look at ?a shows what's going on:

?a

(* Global`a *)

(* a={Nothing,2,3,4,5,6,7,8,9,10} *)

It seems that Part ([[...]]) does not apply the effect of Nothing after the replacement has been done, leaving you with a list that is still 10 elements long. So the second a[[1]]=... simply replaces the Nothing in the first element with Nothing again.

You case use Delete to do the deletion properly:

{a = Range[10], a[[1]], a = Delete[a, 1], a}
(* {{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, 1, {2, 3, 4, 5, 6, 7, 8, 9, 
  10}, {2, 3, 4, 5, 6, 7, 8, 9, 10}} *)

{a[[1]], a = Delete[a, 1], a}
(* {2, {3, 4, 5, 6, 7, 8, 9, 10}, {3, 4, 5, 6, 7, 8, 9, 10}} *)

Part assignment performs in-place modification of an expression without evaluation of the result. At the same time, on the Documentation page for Nothing we read:

Nothing is removed as part of the standard evaluation process.

So after evaluation of a[[1]] = Nothing you still have a List of length 10 with first element being Nothing. You can replace Nothing with anything else in the same way again:

a[[1]] = Nothing;
Definition[a]
a[[1]] = Missing[];
Definition[a]

a={Nothing,2,3,4,5,6,7,8,9,10}

a={Missing[],2,3,4,5,6,7,8,9,10}

You can remove Nothing by evaluating the expression:

a[[1]] = Nothing;
Definition[a]
a = a;
Definition[a]

a={Nothing,2,3,4,5,6,7,8,9,10}
a={2,3,4,5,6,7,8,9,10}

---

Instead of Nothing you can use such functions as Delete, Drop, Take or ReplacePart for the same purpose.