The Rap Dictionary defines the verb drop:
3. To play music, or sing a song, could mean for the dj or mixer to add bass behind the singers vocals. Drop the beat.
and defines the noun beat:
1. Music which is played in the background.
Much of hip-hop has beats played from vinyl records on two turntables, which is rapped over. The instruction to "drop the beat" is from the rapper to the DJ, to start playing a beat or music. Drop probably stems from the physical act of dropping or placing a record on the turntable, or more precisely dropping or placing the needle on the record.
The line "Beastie Boys known to let the beat... mmm, drop!" is from their 1998 track, "Intergalactic":
Tammy D getting biz on the crop
Beastie Boys known to let the beat... mmm, drop
When I wrote graffiti my name was Slop
If you listen from the 2:27 mark, you can hear a beat under the first line, then no beat at all for the next one, and then a new beat comes in after that.
This directly references their own track "The New Style" from their 1986 debut album Licensed To Ill:
Spent some bank - I got a high powered jumbo
Rolled up a wooly and I watched Colombo
Let me clear my throat - Kick it over here baby pop
And let all the fly skimmies, feel the beat... mmm, drop
Coolin' on the corner on a hot summer day
Just me, my posse and M.C.A.
From the 3:06 mark, there's a beat under these first two lines, then no beat under the next two, and again a new beat comes in after the instruction.
To drop a beat can also have an opposite, negative meaning in "traditional" live band music, where player or a drummer misses or loses the beat, stopping the rhythm.