As the old poem says:
For want of a nail the shoe was lost.
For
For want of a shoe the horse was lost.
For
For want of a horse the rider was lost.
For
For want of the rider the message was lost. For
For want of a message the battle was lost.
For
For want of a battle the kingdom was lost.
And
And all for the want of a horseshoe nail.
Exactly the same principle applies in chess. Usually the loss of the pawn results in the loss of the file, loss of control of the file allows the rooks to penetrate and the position soon collapses.