Having lots and lots of weapons. Implies that you could not possibly want any more armaments. May even imply that you have too many weapons. (Of course, we know that is not really possible).
These days we use it figuratively, to refer to anything having lots of something. You can be armed to the teeth with gear of any sort (camping, climbing, photography), or a program can be armed to the teeth with features.
This idiom's first recorded use was in 1849, when it was used by English statesman Richard Cobden in his Speeches on peace, financial reform, colonial reform and other subjects:
"Is there any reason we should be armed to the teeth?"
The phrase "to the teeth" has meant 'well equipped' since the 1300s, where knights could be armored 'to the teeth', i.e. have armor over all their bodies, and also their face! (It was considered pretty fancy at the time). Since Cobden came along with his anti-milatary rhetoric, the phrase 'to the teeth' has rarely appeared in the English language without an 'armed' in front of it.
I assume that this idiom originally referred to someone who had a weapon in each hand, but not happy with that, who also carried a spare knife (or other blade) between his teeth. I have this mental picture of a pirate boarding a ship, literally armed to the teeth. But sadly, I have been unable to find any sources that would describe what mental picture Mr. Cobden may have had in his mind.