Does it sometimes take energy to create a chemical bond?
"Molecules that require less energy to break than they give off when broken are called fuels." is the most imprecise sentence in the paragraph, because fuels don't give off energy when their bonds are broken, but when these broken bonds form new bonds (with oxygen) that are much more stable and release more energy than it took to break the original (carbon-hydrogen) bonds.

A candle is stable in air, even tho it does not have complete stability in an oxygen-containing atmosphere. But if you put sufficient energy into a few bonds, the excited atoms will combine with oxygen, which gives off enough energy to excite a few more bonds, which etc., etc. It's this new reaction, which is overlooked in the paragraph, which provides the increased temperature during the reaction.

The energy that "bonds require to form" is an activation energy which destabilizes a molecule enough so that it can reorganize with other reagents (like O2) to form more stable products. This activation energy may be supplied initially from outside sources (the match), but in a continuing reaction, the exotherm from the reaction provides the activation energy and then some.