1) I BIT THE BULLET and took a flight.
= I took the undesirable/unpleasant action of taking a flight.
* "bite the bullet" = face a difficult/painful situation full-on, or take a hard/risky action. You clench your teeth hard when you experience extreme pain (like cutting the flesh), so to make it easier people may put a piece of cloth (or a bullet if it's in the battlefield) in your mouth to bite down on. So you say something like "I'll have to cut your arm. You bite the bullet and hang on."
2) The majority of carbon emissions from aviation comes from a FEW BINGE FLIERS
= ... from a FEW FLIERS WHO FLY EXTREMELY OFTEN.
* "binge" means an excessive, uncontrolled indulgence in something, like eating or drinking till you drop.
3) As a way of trying to push "GOING FLIGHT-FREE", I don’t really like it (better put the phrase in quotes).
= ... to push THE PRACTICE OF NOT FLYING (AT ALL), I don't ...
* "NOUN-free" means not involving NOUN, e.g. fat-free diet = a diet not involving/including fat.
4) It is particularly important for academics working in climate change TO SET THE EXAMPLE by reducing their flying.
= ... in climate change TO SHOW A MODEL by ...
* "to set an/the example" = to show a good model case for other people to see and follow.