Apply oneself to hard work.
This is a US phrase, although it may well be related to an earlier British phrase 'buckle to', which means much the same thing. That dates back to the early 18th century, for example, from John Arbuthnot's story John Bull in The Law is a bottomless pit, 1712:
"'Squire South buckled too, to assist his friend Nic."
'Buckle down' is first cited in the American literary/cultural magazine Atlantic Monthly, 1865:
"If he would only buckle down to serious study."