to agitate: to stir something into commotion, whether water, a person's heart, or a crowd
At its root this word pictures water being churned so it no longer lies still. That physical image carries every other use: to trouble someone is to stir the calm surface of their heart, and to trouble a crowd is to throw a settled group into turmoil. The New Testament uses it in all three ways, and the same verb ties them together.
The most common use is inward. A person is "troubled" when fear, grief, or alarm breaks their composure: Herod at the news of a rival king, the disciples at a figure walking on the sea, Jesus himself as he faces the cross. It is not a settled mood of sadness but a sudden loss of calm. This is why Jesus can command the opposite, telling his followers not to let their hearts be thrown into that turmoil.
The third use turns outward, to people stirred up in a group. Agitators "trouble" whole cities and congregations, unsettling them with alarming or false words. Here the disturbance is social rather than private, but the picture is the same: what was calm has been shaken into confusion.
A near neighbor, θορυβέω, points specifically to the noise and uproar of an agitated crowd, while this word covers both the churn inside a single heart and the commotion of a mob.