Hesi products will maintain their quality best if they are kept in their original container, properly closed after each use, and stored in a dark and cool (but frost free) place. Something to be especially mindful of, is the effect of direct sunlight on the containers. This can heat up the bottles to a deceptively high temperature, far higher than the general temperature of the room they are in. For this reason, sunlight is best avoided. The garage, the basement, or a pantry will usually provide the desired conditions.
In extremely warm climates, it's actually not a bad idea to store Hesi's bottles in the refrigerator during the hottest days. Just remember, definitely do NOT store them in the freezer, and make sure to keep track of which food is for your plants, and which is for you!