Bhishma is known for his vow that Bhishma will always protect the king of kuru clan. So he was fighting from Duryodhana's side. Bhishma was brahmachari and was strictly performing celibacy. So he was most powerful in Mahabharata war. Arjuna was unable to kill Bhishma because of his feelings and guts for his grand parent.
Seeing this Krishna was in trouble that whose vow should be broken? Krishna's vow of not taking any weapon or Bhishma's vow. So Krishna broke his vow for the sake of Bhishma's lifetime celibacy.
Mene meri pratigya tod di teri pratigya khatar
I broke my vow for the sake of your vow.
Arjuna was stopping Krishna for not breaking vow but could not understand what Krishna was doing. Arjuna stood and promised Krishna that he will fight with full passion now-onwards.
On the other side Bhishma stood on his vow but forgot that one should not protect sinner. Thus Bhishma was fallen by the arrows of Arjuna because Bhishma was cursed that all his sins will become arrow and will pin down his body in his end time.