For example: Altitudes and flight levels:
3,500 is spoken as "three thousand five hundred".
13,500 is spoken as "one three thousand five hundred".
But other numbers (like times, directions, etc) have every digit spoken, for example 10 minutes is "one zero minutes" or wind direction 220 is "wind direction two two zero". Speed reading would be something like "one niner zero knots" for 190 kt.
The sentence structure is also relatively strictly defined, for instance at an unattended airport you would always transmit in this order:
(1) where you are
(2) who you are
(3) what you intend
(4) repeat where you are
So you'd say something like: "Swartkops traffic(1), small Jabiru Kilo India Tango(2), left downwind for runway one eight, touch and go(3), at Swartkops(4)"
This kind of strict structure and limited lexicon does make it understandable after a bit of practice, since you are expecting certain words to show up.
You can read the ICAO Radiotelephony Guide[1], should you be interested.
Searching for something like "ICAO Phraseology Reference Guide" might give you good results too.
[1] http://www.icao.int/safety/acp/Inactive%20working%20groups%2...
The big thing is understanding the vocabulary, and the things they're likely to say. It's called the "pop-out effect" - once you know what you're looking for, it's much easier to find it. You can think of it in Bayesian terms - an increased prior on certain words and phrases makes them much quicker to detect.
(I will also say that many people find American aviators/ATC much harder work. They use less standardised phraseology - perhaps because they don't have so many different native languages all crowded up into a small airspace. See if, eg, Amsterdam or Dublin is easier for you to understand.)
But... not understanding isn't necessarily a bad thing, I can't listen to this and work because I can understand it