This refers to the celestial equator, not the imaginary line that runs around the Earth. It's more or less the same as saying "when the Sun is furthest from the plane drawn through the Earth's equator".
At the equinoxes, the Sun passes through the celestial equator. It then gets progressively further away from it, reaching a maximum at the solstice.
When viewed from the Earth, the Sun is highest in the sky at mid-day on the day of the summer solstice. (This day is also known as the summer solstice.) The winter solstice is when the Sun is at its lowest at mid-day.
(This is the best of my understanding, anyway. If you can explain it better, please let me know.)
Wikipedia expresses it more scientifically: the solstice occurs "when the Sun appears to reach its most northerly or southerly excursion relative to the celestial equator on the celestial sphere."
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