You can get this Achievement very easily, but first you need to learn how to catch a bird :
1 : Find a bird
2 : Hold RB to enter sneak mode and get close to the bird
3 : While in sneak mode, hold A to enter Pounce and use your left stick to adjust the length of your jump, release A to pounce
Note : A "A to Pounce" message will prompt when you're close enough to the birds so you can pounce them.
I suggest training on a few birds so you can build minimal action memory.
When you feel ready (this might still take a few attempts) :
1 : Run/walk in a bird's direction to scare it or wait until it takes off as you sneak too close to it, birds will come back to their take-off position after a few seconds and they won't be scared again if you're in sneak mode when they come back
2 : Aim at the sky and follow the bird and wait for it to start its landing approach
3 : Now make the camera follow the bird and wait for it to reach about a 1,5-2 meters altitude
4 : Adjust the pounce trajectory so it connects to the bird's landing trajectory (use the bird's shadow as a reference point) and release A
If this written solution doesn't help, Void7274Void on TA made a video for you :