Learn English – The meaning of: “Why don’t you just beat it?”

Beat the band is an idiom for to the greatest possible degree.

It's like they're smiling as big as can be.

It's another form of "to beat all", akin to "that beats everything".

From the American Heritage Dictionary in Dictionary.com:

Idioms & Phrases
to beat the band

Also, to beat all. To the greatest possible degree. For example, The baby was crying to beat the band, or *The wind is blowing to beat the band , or *John is dressed up to beat all . This idiom uses beat in the sense of "surpass." The first term may, according to one theory, allude to a desire to arrive before the musicians who led a parade, so as to see the entire event. Another theory holds that it means "make more noise than (and thereby beat) a loud band." [Colloquial; late 1800s]

It has another meaning as well, which is not used in your example:

to beat the band
very briskly; very fast. He's selling computers to beat the band since he started advertising. She worked to beat the band to get ready for this.

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.