Why is 15:00 in London a different day in Sydney?
Because it is: the date changes with the hour, and each row shows the day for exactly that reason.
What is a gap change?
Two places that both move their clocks, but on different dates (Europe in late March and late October, the US in mid March and early November), are a different number of hours apart for a few weeks a year. A meeting set by local time drifts, and the tool names the day.
Why can’t I enter 02:30 on the night the clocks go forward?
Because in that zone 02:30 never happens that night; the tool says so and shows what it read instead. On the autumn night an hour happens twice, and both are named.
How current are the zones?
As current as your browser: the tool reads the browser’s own copy of the tz database. A rule change announced for next year appears when the browser updates, so a far-future answer for a zone that has just changed its rules can lag until then.