In the following King and Pawn ending with White to move, White’s advanced and vulnerable queen side pawns look exposed. However, White’s King appears to be able to advance and protect them before Black’s Marauding Monarch arrives to consume them.
LarsB of Norway(Magnus Carlsen Land) thought that moving his king to d3 would prevent Black King from landing at c4 attacking the weak pawns at b4 and a4. However he has overlooked the exclamation move a5 by Black! Do you see why a5 will lead to the resignation of the first player?
A relatively advanced strategy/technique in chess is to sacrifice material in order create weak squares. In this case, the weakened c5 square creates a path for the Black King to capture White’s indefensible queen side pawns leaving the second player with either a passed a or b pawn and a won position.
Dvoretsky uses the phrase widening the beachhead in the sense of increasing the region controlled by Black in this case. The two intertwined chess element here are material versus mobility or piece activity. In this case, Black gave up material(a pawn) in order to mobilize his King enabling it to attack weak pawns.