Their loss to Olympiakos came 12 months after defeat to West Ham.
Biraghi said: "I've been in Florence for 7 years now, I've been through it all. From being saved on the last day to these two lost European finals. The fans have always followed us, they have made impossible trips and they have never abandoned us. I know many of them personally and I know what the team means to them.
"I understand their disappointment and their bitterness. When Davide died we made a pact, us for them and them for us, and last night, until they left the stadium, I was there.
"Now it's normal for them to be disappointed, we accept it and I accept it. As long as I am captain everyone will give their all to achieve an objective that we have deserved over the years."