The short answer: sometimes immediately, sometimes imperceptibly over time—and in most cases, both! You will almost always notice visible improvement in the tone of your horse's musculature, overall posture, and ease of movement by the end of every massage.
A more nuanced answer depends on the particulars of your horse's needs. Horses who are seen for the restoration of mobility after old injuries will, understandably, require a gradual course of therapy before significant change is noticed. The tissues of horses coming back after acute issues often respond rapidly with massage to facilitate the healing. Your horse's age, condition, receptiveness, and the goal of the treatment all play into to the result of each massage. Because the benefits of massage are holistic and cumulative, a regular, ongoing regimen will yield the most stable results over time.
Many clients find it rewarding to take before-and-after photos on either end of their horses' massages. In this way, they can track both short- and long-term changes to appreciate the value of their investment.