Why are sparks coming out of the bottom of my Tesla coil secondary?
The bottom of the secondary should be connected to some sort of ground connection.

If you are operating indoors, then a large sheet of foil underneath the secondary is ideal. I notice you have a wire mesh cage around your electronics in the base. This could possibly do as an alternative to a metal sheet, if all pieces of it are connected together, but increasing its size with a sheet of foil underneath, connected to it, would be even better. The foil, or your mesh cage, must also be connected to the safety/earth ground of your mains supply at the point where it powers the coil. 

Put a mains filter at the coil inlet. If you don't have a mains filter, use X2 type capacitors to RF-short both live and neutral conductors to safety ground. This keeps RF interference from conducting back into the electrical network differentially between conductors.

If you are operating outdoors, then a spike into the ground is fine.