How can the test pen light up when touching your fingers that doesn't have or have very little electricity current?
What I see in the video is (1) a button battery (coin cell) inside, with access by unscrewing the metal battery cover/cap (using slot type screwdriver at large diameter end of probe), and (2) the fuse continuity measurements that are shown require that you make contact with the battery cap with one finger while holding the fuse-to-be-tested at one end of the fuse and then touching the other end of the fuse to the probe tip.  Thus the "test" shows that current (provided by the coin cell) can flow through your body from one hand to the other and through the fuse to the probe tip. That current is very small and is effectively amplified by circuitry in the device to light the LED using power supplied by the button cell.  The slotted cap (with battery) is best seen at about 1:15 minutes and the contact via both hands is demonstrated at about 1:50 minutes.