Why is fluorine a gas bromine a liquid and iodine a solid at room temperature?

The reason is that the attractive forces between the molecules of these increase from top to bottom of the Group..
The molecules of each substance attract each other through dispersion (London) intermolecular forces.
Whether a substance is a solid liquid or gas depends on the balance between the kinetic energies of the molecules and their intermolecular attractions.
Molecules at the same temp have the same average kinetic energy (##KE = 1/2mv^2##).
In fluorine the electrons are tightly held to the nuclei. The electrons have little chance to wander to one side of the molecule so the London dispersion forces are relatively weak.
At room temperature the fluorine molecules have enough energy to escape these attractive forces. The attractions are not strong enough to make fluorine condense or solidify.
In bromine the electrons are further from the nuclei so it is easier to distort the electron cloud. The London dispersion forces are weak but stronger than in fluorine.
The attractive forces are strong enough to hold the molecules close. But the molecules have enough kinetic energy to slide past each other so bromine condenses to a liquid.
In iodine the electrons are so far from the nuclei that the electron clouds can easily distort. The London dispersion forces are strong.
The molecules no longer have enough kinetic energy to escape the attractive forces. The attractions are strong enough to cause the iodine to solidify.