If lightning were to strike power lines, wouldn't you expect it to increase the power, and hence your room lights get brighter?
Why do they go dim for 1 second at a time?
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Janka comes close, but there are are several more details. (Note, recalling from EE classes about 45 years ago.)
On many high voltage lines there are arc electrodes at various points. When lightning strikes the line, the increased voltage causes an arc to form across the electrodes. This helps to dissipate the voltage of the lightning strike.
But the spacing of the electrodes is such that, once the arc has formed, the normal voltage on the line is sufficient to keep it going. So the electrodes are formed in the classical V shape, close together at the bottom and wider at the top.
Heat causes the arc to rise (see "Jacob's ladder") and so it gets longer. Eventually, when it gets to the top, the voltage will (hopefully) no longer sustain the arc.
If the arc does not extinguish itself, eventually a nearby over-current detector (fuse) will trip and cut off the power.
But to save the lineman a trip to reset it, the over-current detector is often designed with a timer so that it resets itself after a few seconds. But usually there's a limit to how many times it will reset (in case the over-current condition is due to, eg, a downed power line).
Lightning brownout procedure:
There's another explanation that does not require a direct lightning strike.
Storms strong enough to generate lightning also typically have strong winds. A strong gust may bend a tree branch or other vegetation so that it touches a power line and creates a temporary short circuit to ground. This draws a large current, which drops the voltage and causes your lights to dim.
Once the tree bends back, the power is restored to normal, either immediately or after the operation of a recloser, as described in this answer to a related question.
Branches weighed down by snow or ice may be closer to the power lines than in their unweighted position, so it might take less motion of the tree to contact the power line in this condition.