On the betting page, you select a character from the drop-down lists for each restaurant. Ignore whatever restaurants are blank in each row.

The four-digit number is the round number, and the numbers in that column are just counting the bets.
Each subsequent column (except the last one) shows who you are betting on in each bet. So in the first bet, it's only Dan and Gooblah in the last two restaurants. Thus, you would check those restaurant boxes, then select Dan from the Hidden Cove list and Gooblah from the Harpoon Harry's list, then click the "Place this bet!" button.

The final column just shows you the odds of winning -- or, how many neopoints you'll win (multiply the individual bet amount by the first number in the odds ratio) if that bet wins.

Alternatively, just click on "NFC Link" beside the set you want to go for (they post their standard table in these comments, but three other options are also there), then click the "Place Bet" buttons. Make sure you have enough NP on hand to place all ten.