my furnace fan stays on even when at temperature
There is often a "fan" switch on the thermostat, which can be set either to "auto" or to "on" - the "on" setting runs the fan all the time, which can be helpful with distributing heat (or cold) in houses that otherwise tend to be very cold downstairs and very hot upstairs due to natural convection currents, especially if the furnace/AC does not run for quite a while.

It can also be helpful in running more air through the filters. It does use more electricity running the fan - how much depends on the fan.