There's no official guidance on this, so you have to make it up.
The game rules don't provide any description of it feels like to be affected by Aura of Despair - the description of the ability in the Blackguard's class features only describes the mechanics:
Aura of Despair (Su)
Beginning at 3rd level, the blackguard radiates a malign aura that causes enemies within 10 feet of him to take a -2 penalty on all saving throws.
It's possible that somewhere in the expanded D&D canon (e.g., Forgotten Realms novels), someone has written a description of this ability's subjective effects, but I'm not aware of any examples.
That means we have to make something up. Fortunately, the process of fleshing out underspecified game mechanics into emotionally satisfying narratives is something you're probably pretty familiar with - we sometimes call this "roleplaying."
Let's take a look at what little the game does tell us about this ability:
- It's supernatural (that's what the (Su) means)
- It causes you to feel "despair" (or at least that's what the name of the ability says)
- It lowers your saving throws (i.e., makes you less likely to throw off negative effects by being too tough, too quick, or too strong-willed)
To me, that says the overall feeling you get from this aura is probably something along the lines of "I should just give up, everything is hopeless, it's not worth fighting anymore." That lines up with "despair" in the ability name, and neatly explains why the ability makes you less likely to save against hostile effects.
If you're looking for more detail than that, or inspiration for how to roleplay being affected by the aura, you could try looking in fiction for examples of supernatural emotion-affecting abilities. These are actually pretty common in fantasy, sci-fi, and superhero fiction. A couple examples that come to mind for me:
- In J. C. McCrae's Worm and its sequel Ward, several characters have superpowers that affect emotions. This early Ward chapter, for instance, provides a first-person account of a fight against someone whose power causes intense feelings of "loss" - pretty similar to despair. It sounds pretty brutal.
- In Ada Palmer's Terra Ignota series, technology exists to subtly manipulate emotions using chemical agents. SPOILERS:
In the second book of the series, Seven Surrenders, Thisbe Saneer attacks Carlyle Foster by inducing suicidal guilt, despair, and fear; the scene includes a detailed account of Carlyle's subjective experience of the attack.