The DANGER RESULTS table is triggered by non-hit dice on each hero's turn, and you use those same non-hit dice to find the appropriate result on the table.
Every time a hero rolls to Save The Day, count the number of (non-bonus) dice that come up 1, 2, or 3 (the ones that didn't score a hit). If the threat has a strength against the hero's attack, add +1 to that tally for each natural 1 (again, except on bonus dice).
The total number is how much "danger" the hero has rolled. Compare that number to the DANGER RESULTS table to see what happens (in addition to whatever effect the roll's hits have on the threat).
It's not really laid out all at once anywhere, so I pieced this together from the DICE RESULTS section (1-3: danger / 4+: hit --how you roll danger); the BONUS DICE & THREAT DICE section (Bonus dice never count for danger results); the STRENGTHS & WEAKNESSES section (If a threat has a strength against your attack, you take +1 danger for each 1 you roll); and the OUT OF ACTION section (If you roll 6 danger at once, you get taken out! --comparing this to the DANGER RESULTS table is how I knew danger rolls went to the table instead of to, say, the METER).