From my experience, "weak heels" could mean either A.) soft, weak hooves in general that make a horse 'gimpy' on rough terrain, B.) contracted heels, or C.) underrun heels. However, without a picture it's hard to say exactly what your farrier means.
This is a PDF on contracted heels with several images: file:///home/chronos/u-cd8d8a85880d914f2d02fc1ccaa9a0d165ea3a6f/MyFiles/Downloads/contractedheelswebedition.pdf
And if that's not viewable for some reason, here's where you can download it: https://www.researchgate.net/publica...ntracted_heels
This is a decent article that explains a little on underrun heels: https://barehoofcare.com/index.php/h...heals-article/
Movement wise, issues with heels or hoof balance can change a horse's posture a great deal. My mare had unbalanced hooves for a while due to a poor farrier, and in result she moved rather stiff-backed and didn't have good length of stride at all in the front. Fortunately, I managed to find a better farrier and since then she's improved greatly. Likewise, I've had a horse with flat, "pancake" shaped front feet with small heels, and due to this he was often easily sore or bruised if you were riding on rougher terrain/rocky terrain. He never had what I would consider "perfect" feet, but with proper trimming and shoeing they did become much better. I think conformation plays a good deal into the amount of impact on the heels, as a horse with poor/downhill conformation will be heavier on the forehand naturally, and therefore carry more weight on the forehand than a horse with better conformation. Posture can play into it a great deal as well, but often posture is likely more caused by the poor hooves than the posture causing poor hooves.
As for fixing them - well, one way would be through corrective trimming/shoeing, and possibly by some hoof supplements if the hoof itself is weak in general. But I don't think anyone could tell you for sure a way to fix it without pictures/videos. Having a vet look at pictures of your horses feet or the feet in person would also be very helpful to get a second opinion as to what your farrier said, because not all farriers are equal and I typically prefer getting a second or third opinion before venturing to fix something, just in case.