Basically, you just have the work the whole sum backwards.

What he did:
1) give 1/3 away
2) give away 8 cards
3) give away 3/4 of remaining cards
4) give away 2 cards

With 46 cards left,
Add 2.
46+2=48 (this is 1/4 of the remaining cards.) (4)

Meaning, one whole of remaining cards will be :
48 x 4 = 192 (3)

Add back 8 cards:
192 +8 =200 (this is 2/3 of the cards) (2)

Multiply 3/2 to obtain the original number of cards:
200 x 3/2 =300.

There is originally 300 cards.