People sometimes breathe much faster than is healthy. This is called hyperventilation and can be caused e.g. by anxiety or stress. When you hyperventilate, you're saturating your blood with oxygen, and depleting it of carbon dioxide (CO2). CO2 dissolved in liquid (such as blood) makes an acid called carbonic acid. Without CO2 in your blood, you're missing this acid and your blood becomes more alkaline, which causes a syndrome called hypocapnia, which is associated with symptoms such as dizziness, fainting and (more) anxiety.

So, to sum up: people who are hyperventilating run a risk of having too little CO2 in their blood. The idea behind breathing into a paper bag is that you would breathe out air with more CO2 in it (compared to regular air), and then by breathing this back in you would raise your CO2 levels.

There are two reasons why this method is no longer recommended: (1) it's rather ineffective and (2) it can be very dangerous:

1. Experiments have shown that CO2-levels barely increase faster by rebreathing exhaled air, and no more than a placebo condition where people believed they were rebreathing, but were in fact breathing normal air. Why does a placebo method affect CO2 blood levels? Well, because simply relaxing and breathing more slowly is an effective method of raising CO2, and a placebo treatment can help reassure people that they will be okay. They believe that it will help, and that makes them relax.
2. Symptoms resembling those of hyperventilation can be caused by conditions such as asthma attacks or heart attacks. And sometimes people who hyperventilate do so because they are actually low on oxygen (due to some other medical reason). In all these cases, making people breathe into a paper bag can be very dangerous, and even lethal, as you are depriving them of much-needed oxygen.

So the takeaway is that at a minimum, you should exercise an abundance of caution when attempting the paper bag method. Never have someone else hold the bag for you (let alone hold a bag over another person's mouth if you think they are hyperventilating). Never use this method if you suffer from any kind of heart or lung problems, or if you've ever head a stroke, embolism, blood clots, etc. Only breathe into the bag for a few breaths (most sources recommend no more than 6-12 breaths). And avoid using this method if you're not sure that you're hyperventilating.