1893 text
“A game at cards not unlike Loo, but with this difference, the winner of one trick has to put in a double stake, the winner of two tricks a triple stake, and so on. Thus, if six persons are playing, and the general stake is 1_s., suppose A gains the three tricks, he gains 6_s., and has to ‘hand i’ the cap,’ or pool, 4_s. for the next deal. Suppose A gains two tricks and B one, then A gains 4_s. and B 2_s., and A has to stake 3_s. and B 2_s._ for the next deal.” —Hindley’s Tavern Anecdotes. — M. B.
Jenny Doughty Link to this
http://www.davidparlett.co.uk/histocs/loo.html
This excellent link explains the rules of loo, an historic card game of which 'Handycapp' is a variety.
Grahamt Link to this
Hence, if you win one hand, you are "handicapped" by having to put in a bigger stake for the next. The SOED has a different explanation of the game and origin of the phrase, but at about the same date.