A good memory and experience
It's a combination of a few things. Yes, a good memory helps and whilst in the early stages of being a 'new' pilot you are perhaps a little overwhelmed and with a limited ability to multitask, once you become more confident and less overwhelmed with the mechanics of flying and navigation you will have more brainpower available to remember ATC instructions.
Secondly, you will find over time that ATC instructions are very often a standard response and become almost predictable. I know that when I call Heathrow for special VFR they will come back with a long string of instruction. That instruction is more often than not the same each time! Once you've been doing it a while you will see what I mean.
Also, when you call for a basic service, say, and they give you a squark and a QNH, set them on the instruments before reading back. It might delay your response by a few seconds, but you stand more chance of getting it right first time.
Finally, as they say, aviate, navigate, communicate. If you miss what they are saying, ask 'say again'. We're all human and we're not all sky gods. They won't be offended.
I speak from my own experiences!!
Good luck,
Mona