Why does the resistance of a filament lamp increase with current?

It's a metal. More current means more electrons colliding with metal ions in the lattice. This transfers kinetic energy to the ions in the lattice increasing its temperature. This does not cause an increase in the number of delocalized/free electrons as it is a metal. So this increased thermal agitation of the ions in the crystal lattice causes more collision between moving electrons and ions, increasing the resistance.

Hi Tony, filament lamps are usually made of the metal tungsten and will have at low temperatures a constant value of resistance. However as the current increases and the filament becomes brighter the electrons flowing in the wire start to collide with the metallic ions within the metallic lattice structure more often. This affects the flow of the electrons as they attempt to flow through the conductor and the effective resistance will increase. This produces the current voltage graph shown below.

the temperature of the filament gets hot. This because if you increase the current you increase the rate of flow of charge. If you increase the rate of flow of charge more charge can be converted into heat as a by product increasing the resistance of the filament.

Because resistance increases with temperature, and heating in the filament is proportional to the square of current.

As the atoms heat up they 'jiggle' more and this increases the number of collisions that the electrons have with them and this makes it harder for them to pass through the wire and so the resistance is increased