Why is Earth's day getting slightly longer every century?

Answer: The Moon siphons spin energy from Earth via tides

The Moon siphons spin energy from Earth via tides ✓ — Correct! The Moon raises tidal bulges on Earth, and Earth's rotation drags those bulges slightly ahead of the Moon. The misalignment tugs back on Earth (braking its spin) and forward on the Moon (pushing its orbit outward ~3.8 cm/year). Net result: our day lengthens about 1.8 ms per century.

The Sun's gravity slowly saps Earth's rotation — Not quite. The Sun does raise tides on Earth, but the solar tide is only about 46% as strong as the lunar tide. It contributes, but the dominant braker by far is the Moon.

Space dust keeps landing on Earth, adding mass — Not quite. Earth does accrete roughly 40,000 tonnes of space dust every year — but Earth's total mass is 6×10²¹ tonnes. The dust is a rounding error billions of times too small to measurably slow the spin.