This song may help you remember a basic pronunciation rule of English: Words ending in -MB don't pronounce the letter B (even when they add a suffix).

Climb /klaɪm/ Climbing /klaɪmɪŋ/
Bomb /bɒm/  Bombing /bɒmɪŋ/
Lamb /læm/

30 DEGREES= The inclination of the plane (so they are falling down).

STALLING OUT= When an engine or mechanical device stalls, it suddenly stops working.

BEACON= A radio transmitter that emits a characteristic guidance signal for aircrafts. The beacon is at the airport, so they are still 24 miles away from the safety of the airport.

CRACK= A gap, an opening. In this case it may refer to a lightening, which renders the sky like a crack.

THE WARNING'S OUT= The alarm has been activated. Danger is coming, I (or they) am warning you.

DIVE= If a person dives into the water, they jump into it head first. If you take A DIVE, you jump into the water head first. If a plain takes a dive, it suddenly drops and starts quickly falling down, head first.

PUSH THROUGH= Move through, but with difficulty (as if you need to push to be able to move through that).

FIVE MILES OUT= They were five miles out of their way because the strong hurricane has pushed them out of their route, so now they are in danger and the radars can't find them.

BAND OF RAIN= A long area of rain which looks like a curtain. When an area of cold air meets an area of warm air with a lot of humidity, the water in the warm air condenses due to a drop in the temperature (colder air can contain less humidity) and heavy rain takes place in the area where the two masses of air meet, creating a band of rain all along the border.
When he says that the plain needs to "push through that band of rain", we get the idea that the curtain of rain is so heavy, so thick, that it will need a big effort to cross it.

HOLD YOUR HEADING TRUE= Your heading is your direction, the place where you are heading (if you head for a place, you go there); to hold your heading is to keep your direction without changing your route. To hold your heading true (?) I guess it means that he must keep his route correct, without any diversion (he's addressing the pilot).

GOT TO GET= (coll.) You've got to get, you must get.

YOUR FINEST= The best of you.
The phrase "got to get your finest out" means: you have to do your best (to get something out is to show it)

YOU'RE NUMBER 1= You are the best. These are just words of praise to give the pilot a sense of courage and confidence so he can deal with the situation better. But here is another interpretation given by Buckler (USA), which makes much more sense to me now (thanks Buckler): As to the phrase "You're number one, anticipating you", I always assumed that to mean that air traffic control had cleared the plane in question for immediate landing, while holding off all others in the traffic pattern, thereby making it "number one" in priority for landing. It may hold a double meaning as a confidence-builder, though.

CLIMBING OUT= To climb is to go up, to rise (you can climb a mountain), though if you climb down, you do the opposite, you descend, go down (tell the baby to climb down the chair, it's too high for him). To CLIMB OUT would be "exit by clambering" or "by ascending", like if you climb out the swimming pool. But talking about what the plane is doing (cause he's still talking to the pilot) he probably expresses the idea that the plain, trapped in the storm and the heavy band of rain, must make an effort to escape the storm by flying up and out over it. Later he will use "climbing out" meaning "to get off the plane"

MAYDAY= In air and sea international jargon, "mayday" means S.O.S = Help!!!
It's really a funny cry for help since it is the English corruption of the French phrase "venez m'aider" (come to help me), and "m'aider" with an English accent sounds "mayday"; so nothing to do with any bright day of May :)

CALLING ALL STATIONS= You say this when you're trying to contact all possible radio stations (because you're asking for help).

GOLF MIKE OSCAR VICTOR JULIET= International code to spell a word avoiding confusion, where every word represents a letter (its initial letter), so this gives us: GMOVJ, which is like the plane's licence number. Look at the painting on the album, you can actually see the letters on the plane. This is a callsign asigned to Scotland (The two first letters (GM) say that). Radio Callsigns prefix are internationally established and assigned each one to one country.

IMC CU.NIMB.ICING= This code represents the rest of the emergency message the pilot is transmitting and goes like this:
IMC= "Instrument Meteorological Conditions", meaning that the weather is so bad that he can't see anything and has to fly by instruments only. This is because he's flying through:
CU.NIMB, which means "cumulonimbus", which are bad-weather clouds, bringing lots of turbulence, rain, hail.
And ICING: Ice is getting on the wings and propeller blades of the plane, which is extremely dangerous because it will disturb their aerodynamic shape. With enough
icing, the plane is sure to go down.

IN GREAT DIFFICULTY= In big danger.

OVER= We say "over" when we finish our radio communication and we change from the transmitting mode to the receiving mode, so other people know we have finished talking and now they can change to their transmitting mode and speak to us.

THE TRAFFIC CONTROLLER= The person who is at the traffic control tower of the airport helping planes to land, take-off or just keep their route.

VICTOR JULIET= As we said before, this is VJ (part of the plain's identification code).

YOUR IDENTITY= The traffic controller is asking the plain to identify.

I HAVE LOST YOU= I have lost your signal. Since it is the traffic controller talking, this means "I don't know where you are anymore, I can't see you or identify you on the radar".

SQUAWK EMERGENCY= Airplanes are required nowadays to carry a small radio transmitter which transmits a small amount of airplane information to control towers and such so that they can identify those strange blips on their radar screens. When they "squawk", they send this information out. "Squawk emergency" means to send this information on a special emergency band so on the radar this plane will appear clearly as sending an SOS.

LOST IN STATIC 18= If you look at the cover of Five Miles Out, the plane is a Beech 18. If your radio is breaking up, the control tower would tell you "Lost in static" and then give the call-sign, so it was "Lost in static, 18.

CLOSING IN= If the storm is closing it, it is getting thicker, stronger and is covering up all the sky and closing us inside it.

GOT TO PUSH THROUGH= You have to fly through it (see "push through" above).

TRAPPED IN LIVING HELL= We are trapped in a horrible situation. "Hell" can be used metaphorically to describe a terrible situation, and "living hell" is a more emphatic expression.

PROPELLER BLADES= To propel is to cause to move forward. The propeller is the engine that makes the plane move (two or 4 engines placed under the wings), and the blades are the flat pieces that move the air when the engine is spinning round.

STILL= Not moving, stopped.

THE EVIL EYE= The eye of a hurricane is the centre of it. He says the eye is "evil" (bad, wicked) because it is very dangerous for them, and "is coming in now for the kill"= is coming to kill us.

RIDER IN THE BLUE= The pilot. He calls him "rider in the blue" in a flattering way because he is comparing him to a soldier riding a horse, but the pilot is not riding on a horse, he's riding in the sky, "the blue".

BRAKING= To brake is to press a pedal, button, etc. to make a machine (car, etc.) stop or slow down. A plane brakes when it is landing and needs to stop (safe at last).

CLIMBING OUT= If you climb out of the plane, you go out of it down the stairs on to the safe ground.