To load up is to play at the highest stake a game allows. Slot software caps how much you can bet, and on many slots the top stake combines the highest coin value with the most coins per line on every payline. A player betting at that level is loading up, often with a single press of the Bet Max button.
The term comes mostly from slot players, on land-based machines and online video slots alike. You may also hear it at other games for a player betting the maximum, and the meaning is the same. Players load up for different reasons. On some older machines and certain progressive slots, the top jackpot is only available at maximum coins, which makes loading up a requirement rather than a choice. Operators should state in the paytable when a feature depends on stake size.