Look at where the composition actually points. Every figure in the fresco's lower register, whatever else they are doing, orients around a small object on a modest altar, a monstrance holding the consecrated host. Popes and Church Fathers debate, gesture, consult texts and each other, but the host is the fixed point everything else circles, the single physical object connecting the crowded, arguing earth below to the serene, settled heaven above.
That composition is itself a theological claim, made in paint rather than in words. The doctrine of the real presence holds that the bread consecrated at the altar is not a symbol standing in for something absent, a reminder of a distant historical event, but a genuine and continuing point of contact between the material world and the divine. Raphael gives that abstract claim a visual argument. Draw a line from the dove of the Holy Spirit through Christ enthroned in glory down to the host on the altar, and the line does not break. Heaven and the object on the altar occupy, in this fresco's own logic, a single continuous reality.
Centuries of theological argument about how, exactly, that presence works have since divided Christian communities in ways this fresco does not attempt to settle. What it does is show, more directly than words easily can, why the question was ever considered worth arguing about at all.