solder wire makes solid & glassy residue
There are solders with no-clean flux that you can leave on, but the solder you linked to looks like it has the conventional kind of flux core which should be cleaned off.

You can use a spray can of flux cleaner and a small brush (a toothbrush works great) or you can just use some 90%+ isopropyl alcohol in place of the spray can.

The flux isn't conductive, but can cause long-term problems if left on.  The glassy residue is just excess flux, and the "greasy" substance is likely burned flux and condensed water vapor.