(THE FOLLOWING GLITCH HAS BEEN FIXED as of July 29th 2015 so the note is no longer relevant. First of all, you may not be able to obtain this achievement due to a glitch causing one of the dinosaurs to disappear - you may get the gold brick for healing it, but it will not count as healed and the map will still be showing it - this glitch has been observed both on XOne and PS4. If it appears, you can hope that TT games will patch the game OR you can start a new game.... in which case you would have to do all the story levels again to unlock freeplay... end even then, who knows, the glitch might appear again.)
You need to heal 10 dinosaurs in freeplay - in the hub worlds. (Dinosaurs healed as a part of the story don't count).
While doing story levels, always open information kiosks you run across in between levels. The kiosks add the location of red/gold bricks, workers in peril and dinos to heal. If you've opened all the kiosks, going for this achievement later on will be so much easier.
Here's the location for all the sick dinos:
JURASSIC PARK I - 3 sick dinos
Gallimimus Territory
Dilophosaurus Territory
Triceratops Territory
THE LOST WORLD - 2 sick dinos
Long Grass
Hunting Plains
JURASSIC PARK III - 1 sick dino
Ankylosaurus Territory
JURASSIC WORLD - 4 sick dinos
East Boardwalk
Safari Plains*
Shady Grove
Gyrosphere Valley
* some may observe a kind of a glitch here.... if the said dinosaur is not there, you will need to get the gold brick located in the same area first. Thanks to Momie for this tip. Momie says: Get the gold brick ( you need to kill a bunch of small dinos ), fast travel to another island (to trigger a load), go back to the area and the dino to heal will be there.
There is a counter that shows you how many dinos you've healed.
The actual healing should not present any problem... for each one you need examine the dino's dung which will reveal three things you need to get. Getting the things is a mix of typical LEGO minipuzzles. Break stuff, use special abilities of the characters you have etc.