The centre is part of the Refugees as Survivors New Zealand which is a mental health agency for all incoming United Nations refugees.
Many of them have experienced torture or trauma and are unsure how to start their new lives.
They come from all over the world - including Ethiopia, Somalia, Afghanistan, Iraq and Syria - and Ms Swarbrick said they were used to having nothing.
"A lot of people think when they come here all their problems will be solved, but going to a different country, where you don't speak the language, understand the culture or know how you will settle in is very stressful."
More than 750 refugees have been arriving at the centre every year since the 1970s, with intakes of 130-150 people coming every eight weeks.
Faysal Ibro is a 30-year-old Ethiopian student who caught the attention of the United Nations when he was imprisoned in Thailand for having a fake passport.
His jail cell had 30 other people, one toilet and the smell was so bad the guard would fling food at the prisoners from a distance.
Ms Swarbrick said he asked her for a bike because he wanted to feel the freedom of being able to ride around.
"Nowadays he's always wearing a smile on his face."
Auckland Airport spokesperson Charles Spillane said the airport chose the centre because the bikes created a practical way to make a difference in the resettlement of Auckland Refugees.
The $120,000 comes from donations by travelling members of the public and the Auckland Airport.
A Herald-Auckland Airport charity project: Auckland Airport is giving away $120,000 to NZ charities as part of its 12 Days of Christmas initiative this year. The Herald is profiling each of the winning charities.