|When we apply a voltage to a wire, the conduction electrons drift through the wire - forming a current.
Now lets concentrate on a few electrons. Each time an electron collides with an ion, it loses energy. This energy is transferred to the ion (which vibrates more); eventually the energy from billions of collisions is shared amongst all the atoms, and the material gets hot.
This is often called joule heating, and it is something electrical engineers try to reduce unless they are designing a heater!
Once the ions are vibrating, they hinder the flow of the electrons. This is because the electrons will be scattered off the vibrating ions.
As the wire gets hotter, the ions vibrate more vigorously. This makes it more difficult for electrons to pass along the wire. Hence the resistance increases.