Hot air rises; this action brings new air, with more oxygen to the Fire. Hot air rising can also carry burning embers to other areas. Staircases, etc. within a building can act as chimneys to help spread the fire.
This is the transfer of heat through a material, remember how the saucepan handle gets hot. Metal pipes, ducts, etc. can become conductors of heat and spread the fire to materials touching these items.
Remember how that electric fire radiates the heat across the room, or that you can’t get too close to that bonfire. Fuel close to the fire can get hot enough to burn without actually touching the fire.
As the fire spreads it can come into contact with more fuel, which can be waste left lying around, poorly stored chemicals, etc.
Smoke spreads much faster and over a greater distance than fire; this is why it is so important to keep doors especially fire doors closed.