Looking at the history of the phrase in Google books, "Do the best you can" was a quite common phrase in the 1700s (although because of the long s, you have to search for "beft" to find it). After "do", "the best you can" is a noun phrase (e.g., "the best job you can"). People seem to have turned it into an adverbial phrase by using "as", giving
try to explain as best you can.
I believe that strictly, this may not be quite correct grammar, but the alternative
try to explain the best you can,
don't seem quite grammatically right to me either. Both of these forms have been present since the late 1700s, and both are still used today.
Certainly, "the best you can" doesn't work in some situations with transitive verbs:
*Procure the best you can two horses ...
Procure as best you can two horses ... ,
which may be the reason that both forms are still in use.