(To), in races, etc., means to equalise the weight of two or more riders by adding weights to the lighter ones, till both (or all) the riders are made of one uniform weight.
“He carries weight! he rides a race! `Tis for a thousand pounds.”
Cowper: John Gilpin.
To carry weight. To have influence.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894
More on Carry Weight from Fact Monster: