Short answers, pulled from the story.
The frost line, also called the snow line or ice line, is the minimum distance from a young star where temperatures drop low enough for volatiles such as water, ammonia, methane, carbon dioxide and carbon monoxide to freeze into solid grains. Beyond this line those grains can accrete into gas giants and ice giants, while within it only heavier material forms smaller rocky planets.
Estimates for the water frost line's distance vary by model, spanning roughly 2.7-3.2 astronomical units. Hayashi calculated 170 K at 2.7 AU in 1981, Podolak and Zucker found 143 K at 3.2 AU to 150 K at 3 AU in 2010, and Martin and Livio put it at 3.1 AU in 2012.
Beyond the frost line, lower temperatures let far more solid material condense, giving planetesimals enough mass to accrete into gas giants and ice giants. Within the frost line, only heavier, less volatile compounds are available, which is why the inner Solar System formed smaller rocky planets.
The formation frost line, in effect during the Solar System's early history, sat at about 5 astronomical units, while the current frost line marks where water ice can remain stable today. The two differ because the early solar nebula was an opaque cloud with lower temperatures near the Sun, and the young Sun itself was less energetic than it is now.
The outer asteroid belt contains icy C-class objects, while the inner asteroid belt is largely devoid of water, implying the frost line sat at around 2.7 astronomical units during planetesimal formation. The dwarf planet Ceres, at 2.77 AU, lies almost exactly at that estimate and appears to have an icy mantle, and water ice has been detected on 24 Themis at 3.1 AU.
Hot Jupiters are giant planets discovered orbiting well inside their star's frost line, even though giant planets are thought to require conditions found only beyond that line. Astronomers think they originally formed outside the frost line and later migrated inward to their current close orbits.