For the sake of an example, assume I'm a level 1 Sorcerer or Wizard who's somehow had the misfortune of winding up with 1 CON. Maximum HP at level 1 is determined by the highest possible roll on your hit die plus your CON modifier. For me, that's \$6+(-5)=1\$. Horribly low, but technically survivable. I just have to not get hit.
The problem arises when I level up. For simplicity, let's say my group doesn't roll hit dice to determine how HP changes. We instead use the alternate option of adding the listed flat values from the class descriptions in the Player's Handbook (the average roll on your hit die, rounded up). Using this method, the formula for what's added to my maximum HP is: $$\left (\frac 6 2 +1 \right)+(-5) = 4-5 = -1$$
Uh oh. At level 2 my maximum HP is 0. If I somehow manage to keep gaining levels, the situation becomes even more dire; the reduction of one maximum HP stays constant, so as I level up the most health I'm allowed to have goes deeper and deeper into the negatives.
Obviously this is a pretty contrived example, but it could come up in the future. As of now, I'm only aware of two sources of ability damage in 5e, and they hit INT and STR. But if a future book adds a way for your CON to be lowered, it could be disastrous. When your CON modifier is modified by 1, your maximum HP is affected by 1 point for each of your levels. The way the math works out, this means that if you want to calculate a character's maximum HP, you can assume that they've had the same CON modifier since level one; the result is identical. If I'm a level 20 Sorcerer or Wizard and my CON is somehow reduced to 1, my maximum HP immediately becomes -18!
...Is this really how it works? Or is there a rule saying (for example) that your maximum HP will always be at least one, preventing the issue?