How many people do you know in your life who pretend to be something that they're not?
Maybe they're trying to make their parents proud. Maybe they're pretending they're not hurt by something so that no one sees their "weaknesses".
And maybe they've squashed all their passions, all their life's desires to meet what they think society expects of them.
These people are not being TRUE to themselves. They're ignoring what they would TRULY want because they fear the fallout should they pursue it.
But that's not all that the quote is about...
"This above all: to thine own self be true, and it must follow, as the night the day, thou canst not then be false to any man."
That is the full quote. When people ignore what their heart tells them, they deceive themselves. And because they do not know better, they believe the deceit to be truth... and therefore wind up lying to everyone.
So if one is true to themselves, it follows (just like night must follow day) that one will not lie to others.
The line is from Hamlet, a beautiful play.