The Quick Version: You transmit a message with your radio, the repeater receives the message, and the repeater simultaneously re-broadcasts that message, usually with much more power and range.
Now for the some interesting details…
Many repeaters listen for a special tone, which is included in the signal that carries your voice when you transmit your spoken words. Usually this is programmed into your radio when you set up the frequency to use with the repeater. Without that special tone, the repeater won’t repeat what you transmitted.
The offset tells your radio what distance to move up or down the frequency spectrum, in order to order to match what the repeater will receive and transmit.
For example, a repeater will receive a signal on 146.050 MegaHerz (MHz), and then re-transmit that same signal on 146.650 MHz (with a lot more power and range, so that many more people can hear it).
This means that when you use your radio, you will press the transmit button, your radio will transmit your voice on 146.050 MHz, the repeater will receive your transmission, and then it will re-transmit it on 146.650 MHz. That way your radio can both broadcast and hear any replies, but now instead of having limited power and range (since in this case, you may be using a low-power handheld radio), now your message can be broadcast from the top of a nearby mountain (where repeaters are often located) with many times the power. Cool, right?