Notice this:
1 - The first time Bertold and Reiner were attacked by mikasa, this yellow electricity started coming out them (like pouring out) before their transformation into titans. No lightning bolt was actualy seen, falling from the sky and hiting any of them (Eren later included).
The anime shows that the sun was clouded at that PARTICULAR time and Common Titans enter in hybernation if unexposed to sunlight, meaning that the sun works somehow like a energy source to them (speculation).
2 - Now, the fact that most people seen to forget about lightnings is that a lightning traveling through open air emits white light, but can appear in different colors depending on local atmospheric conditions - given there is enough distance between the lightning and the observer. Sooooo, shouldnt it be white, both the bolt and the electricity that comes out of Bertold, Reiner and Eren? Why is it yellow like the sun? Also, i think the lightning only apears when the sun is up and exposed. Recent battles on the Manga are happening at night and they DONT SHOW ANY LIGHTNING on any of the transformations. Its just my opinion but i think that the yellow lighting bolt falling, has something to do with the sun.
3 - Lets remind that the writer of Attack on Titan sees the anime as the "Final Product", not the manga. This actually means that the anime is more accurate to his art than the manga itself, wich is in black and white.
4 - In recent episodes of the anime Eren used the Coordinate Ability. And when he did it, did you guys notice something...? The yellow electricity/lightning appears again, quickly passing through all titans shifters arround (and in my opinion, the common titans as well).
[MANGA SPOILER ALERT] It is said in the manga that the coordinate ability powers are based on the fact that the Eldians (titan shifters and people that can become titans, for those who dont read the manga) are all conected by invisible lines that are out of space (and time in my opinion) and the coordinate ability puts the shifter on a "spot" that intersect whith ALL THE LINES (not like the center, but the only point where all the lines, that pass through everyone, intersect). Thats also explain the fact that if a titan shifter dies without passing the power ahead, the power is not lost, it goes to a ramdom Eldian.
Resuming: i think the yellow lightning bolt/electricity represents the connection between all Eldian people. Other than that, i have no idea where it comes from, but is not some simple scientific explanation, thats for shure