Tell us about the heating unit, first.
Sounds like you have oil-fired forced-air for heat and gas for domestic
Assuming that, hearing "blower kick in" would seem to indicate that the
furnace temp got up to fan-switch set-point, because the burner was on.
If ... you had the thermostat set to heat with fan on automatic.
For a bunch of reasons, I'd call the oil company, and have them perform
normal annual service on the whole heating system. Many bad things can
happen with oil-burner out of "tune."