What's the right sentence and why?
Before letting me understand.
Before let me understand.
If you click here you can see that the second one is the most used.
As far as I know every time when you have BEFORE/AFTER you must use ING.
Those two sentences have different meanings.
First things first, take a look at sentence number one:
Before letting me understand, (...)
You want the person to explain it to you, but not yet: First, you want something else.
The second sentence means the opposite:
(...) Before, let me understand.
The person is about to do something, but before the person does so, you want an explanation.
Your Google Fights is not very meaningful, because the number of results is very low (for more detail, ask on Statistics.SE). Also, it is unlikely that websites use these exact sentences.
As far as I know every time you have BEFORE/AFTER, you must use ING.
Example: (sign in restaurant bathroom) Employees must wash hands after using the bathroom.
Your example phrases aren't very helpful, because they're so fragmentary and artificial-sounding. I will rewrite @FNek's example "Before, let me understand" to "Before, let me make sure I understood you right."
I would not say this. I would say, "First, let me make sure I understood you right."