The mechanical consequence of allowing a Bard to use Bardic Inspiration on themselves is that, starting from 1st level, in addition to the other class features, the bard can exchange one use of their Bardic Inspiration in order to talk to themselves as a bonus action and grant themselves an Inspiration die.
That is literally the only mechanical consequence.
Does that unbalance the game? That might be the actual question you are trying to ask. If that is the case, the answer will be subjective.
If you must have some sort of reason of why it shouldn't work, I will throw two:
It dilutes the role of the Bard in the party.
It would also imply a logical contradiction, see below.
Bardic Inspiration implies that the words used provide some kind of information (praise, encouragement, hints, etc) which helps the ally with their task. Bards know how to convey this information in a manner that is useful for their allies (thus, this is a class specific feature).
Nothing in the wording of the feature suggests that the words are magical. Otherwise, Bardic Inspiration wouldn't work inside an antimagic zone or similar.
Bardic Inspiration requires the words to be spoken aloud for the ally to hear. If you were to modify it for use on oneself, you have to answer this question: Does the Bard have to speak aloud to himself to get the benefit? Can he silently use this feature simply by thinking?
Also note that the number of times you can use this feature depends on your Charisma, thus we can assume it depends on your strength of personality / social skills.
If you allow the Bard to use his Inspiration without talking to himself, this would be drastically changing the way the feature works. It would remove a weakness that Bards tend to have (the need to speak/sing/dance to perform many of their tricks not directly related to hitting things). Not game breaking, but definitely a larger impact. It would also suggest that Bardic inspiration is some kind of psychic or magic power, since the Bard already knows all the information and doesn't even have to do any sort of Bard thing (spout lore, dance, sing, play an instrument) to convey it.
Finally, if you decide to keep the prerequisite of having the Bard sing/praise/explain stuff to himself, balance is more or less preserved, but it would look pretty silly. Other characters might be unable to take such a Bard seriously. And why would such an ability be limited by their CHA modifier?