Offset primarily affects your mechanical trail - that is, the distance between the imaginary straight line going through your head tube down to the ground, and your tire’s contact patch. Normally, the contact patch is behind the steering axis, hence the nomenclature “trail”.
Image from Wikipedia
Increasing the fork offset will bring the contact patch forwards, so trail is reduced.
Decreasing the fork offset will bring the contact patch rearwards, so trail is increased.
Trail affects the steering, as it is basically a measurement of the amount of influence your steering inputs actually have on the contact patch.
Also related are the effects of wheel flop generated by your head tube angle and fork. Since your HTA is (hopefully) not vertical (90°), your wheel will pivot side-to-side when you turn the bars, which adds stability. Among other reasons, this is why head tube angles get slacker as MTBs get more aggressive. Furthermore, fork offset is measured perpendicular to the steering axis. Since your steering axis is angled because of the head tube angle, the fork's offset actually has both a vertical and horizontal component referenced against the ground. The vertical component exaggerates the influence of wheel flop on the steering.