In: Physics
The reason for such is due to difference in pressure of the fluid.
In case of bag, it is hung above the patients arms because to inject the fluid inside the patients arms, the pressure of the fluid must be greater than the pressure inside the veins of the arm of patient. In absence of any external source of pressure the fluid pressure in bag is to only due to height given by gh. So when the bag is at the level of patient arm then h=0 with respect to the position of the patient arm. This results in zero pressure of fluid with respect to pressure inside the patient arm. Hence to create that pressure difference, the bag is hung at some height.
In case of syringe there is a pump that acts as external source of pressure. Hence pump causes increase in pressure of fluid within the syringe as compared to the pressure inside the patient arm. So even at horizontal level when the pressure of syringe is greater than pressure of patient arm then also the fluid can enter into the patient arm.