For circuit analysis, we don't really need fields. A simple one-dimensional description is perfectly sufficient.
Let's first lock at the simple case of a battery connected to a (moderate) conductor. A conductor is a material where the electrons are "loose enough" so that they can move without too much force required.
When the battery touches the ends of the conductor a force proportional to the voltage is applied to electrons and they start to accelerate. However, there is always friction inside the material (ignoring superconductors) so the electrons accelerate until the force of voltage and frictions balance each other. If they go any slower, the voltage will accelerate them and if they go any faster the friction will slow them down.
Friction turns kinetic energy into heat and that's exactly where the energy goes. The friction of the electron flow generates heat. The amount of energy is a function of the "electric friction" (resistance), the driving force (voltage) and the amount of electron flow (current). 395
So we have
$$P = VI = V^2/R = I^2R$$
where $I$ is current, $V$ is Voltage , $R$ is the resistance, and $P$ is the dissipated power.
Now let's look at a battery, two wires, and a light bulb. A "wire" is a conductor with a very, very low friction. Electrons can accelerate freely and nothing slows them down. Hence it's a REALLY BAD IDEA to put a wire across a batter since things tend to melt, explode, burn or do other nasty things.
The light bulb has a much higher resistance than the wires so the light bulb determines the speed limit for the electrons. All electrons go at the same speed since there are sitting "one behind each other" and the electrons in the wire can't go any faster. That means that almost all the energy is dissipated in the light bulb. The bulb and the wires see the same current but the resistance is vastly different since. If the resistance of the bulb is 1000 times higher than that of the wire, it will consume 99.9% of the entire energy.
In short: for regular resistive materials all the energy goes into heat that's generated by electrons bumping into things. 395