I feel like this should be self-explanatory, but I'm not quite sure how it works. Is it essentially a secondary DC that is 5 less than the initial DC? I can not seem to find an example of it anywhere, so one would be appreciated.
From a plain English reading, failing by 5 or more means having a saving throw roll that is a number 5 below the normal DC or less (see detailed explanation below). This does nothing in general but certain abilities care about how much you failed your save by. For example, the pseudodragon's sting (emphasis mine):
(..) and the target must succeed on a DC 11 Constitution saving throw or become poisoned for 1 hour. If the saving throw fails by 5 or more, the target falls unconscious for the same duration, or until it takes damage or another creature uses an action to shake it awake.
In this case, if a creature hit by the Sting rolls a Constitution saving throw. Then, after applying all modifiers to the save, if the saving throw value is...
- 11 or more: nothing else happens
- between 7 and 10: the target is poisoned
- 6 or less: the target is poisoned and falls unconscious
WARNING: This may be all you need to know, the rest is a very detailed explanation.
Why does failing by 5 or more mean having a number 5 below the DC or less?1
In D&D 5e, in the basic rules introduction we have an explanation on what rolling a saving throw entails. First you roll a d20 then add any applicable modifiers. The results are determined as such:
If the total equals or exceeds the target number, the ability check, attack roll, or saving throw is a success. Otherwise, it’s a failure.
The target number for an ability check or a saving throw is called a Difficulty Class (DC)
So for the case of a saving throw, we have a fixed number, called a DC, and the saving throw succeeds if the result of the roll (including modifiers) equals or exceeds the DC.
Failing by a number certain is never defined so we have to resort to a normal English interpretation of what this means. Usually if one says "I failed by 1" it means "If only I had got a result that had a difference of 1 from what I got, I would have succeeded".
In the case of a saving throw, the only result that would satisfy this condition would be a result that was exactly 1 less than the DC. In the example of the sting above, the DC being 11 means that someone that rolled a 10 could say "I failed by 1" or, in other words "If only I had got a result of 1 more than what I got (11) I would have succeeded.
In a similar fashion we can interpret "failing by 2" to mean "having a result of a saving throw that is 2 below the DC" and so on, so that "failing by N" means "having a result that is N below the DC".
"Failing by more than one" usually means "failing by a number that is greater than 1" and so we can also similarly interpreted it as "having a result that is 1 below the DC or less".
So we can conclude that "failing by N or more", in the case of saving throws in D&D 5e should mean "having a saving throw result that is N below the DC or less".
So, in the example of the sting, "failing the result by 5" means "having a result that is 5 below the DC of 11 or less". Since 11-5=6 then this would happen when having a result below 6, as was the case in my example above.
1. Detailed explanation suggested by Yakk
Your understanding is correct. There is no keyword or special meaning at work here (and when something isn't defined by the game, standard English is applied to understand it), to fail a DC by 5 or more simply means that you roll a result 5 (or more) below the DC.
Example: The DC is 15. I roll an 10. 15-10 = 5, I have failed by 5 or more.
Another: DC is 15. I roll 11. 15-11 = 4, I have failed, but not by 5 or more.
Note the "roll" in those examples would be to total after adding modifiers, so if I have +3 on the roll, the first example I would have rolled a 7 (7+3=10).