The effect is described in the on-screen manual as "Attack base value +2 for first attack to a ground or sea target after start fron city, carrier, or military base". That seems to mean that you get the effect of two extra weapons modules for the first shot. Example: if your module strength has reached 51 (the usual after all non-future advances), you get an attack strength 102 higher than what your actual weapons modules amount to. (You can see the "102" just before the Bomb icon in your unit info popup.) You could have six weapons modules totalling 306 (diminishing, of course, with each shot in proportion to your remaining health), or you could reduce to five weapons modules but add the Bombs module, so that your first shot is worth 357 and subsequent shots are worth 255 (factored down for health reductions).
One situation where that attribute could be worth the extra cost and reduced basic attack strength could be where an enemy horde is approaching with a mixture of defensive units and unprotected fast attacking units: your first strike would be at the nearest defensive unit, followed by weaker shots at as many fast attackers as you can manage before fuel or health runs out. Using the "306 v 357" example above, with the Bombs option you could safely shoot any enemy unit with defensive strength less than about 356 - or less than 178 if it's a Fanatic - then, with health diminished much less than if your strength had been only 306, you may be able to hit at least as many poorly-defended fast units as you could have under the "all-306" regime, or maybe more, or the same number but at less cost to your health. Then retreat to recuperation and get back to 100% health faster.
The above examples assume no airport or other experience enhancement. It would be a rare situation where you tried to send an expensive bomber out without having made the small investment necessary to build an airport and hence give the plane two experience stripes right from the start, adding 50% to all of the above numbers. Start building airports as soon as you start researching Advanced Flight if not sooner!