I've seen many chess games, where after a certain point, one side maneuvers their pieces (like doubling the rooks) to target a single point. Most of time it's a pawn on the wing and the other side tries to defend their pawn. But why struggle for a pawn?
As the old poem says:
For want of a nail the shoe was lost.
For want of a shoe the horse was lost.
For want of a horse the rider was lost.
For want of the rider the message was lost. For want of a message the battle was lost.
For want of a battle the kingdom was lost.
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.
I'm not at all disagreeing with the existing answers - both sound. Indeed every pawn is a potential queen. However, one aspect of the question remaining is: why struggle for a pawn as opposed to a more decisive plan?
In the games that you are watching, if they are between capable and well-matched players, very often the game will be quite well balanced with neither side having a promising attack leading directly to checkmate. Launching such an attack in a position where it is not justified is more likely to lead to a worse position than an improvement - an unfounded attack is usually beaten off by a competent defender, leaving the attacker with a compromised position.
But a player has to try something to win and, in a fairly level position, the win of a pawn - even if not conclusive - may be the best plan available to get an advantage in the game. Even a small advantage is better than none!
Another point which has not been raised by the excellent answers above is that pawns are easy to block. A pawn can be blocked by just one piece in front of it, and it cannot take the piece which blocks it. Contrast this with any other piece, which can take a piece which blocks its movement (barring other circumstances such as a pin).
This property means that it is much easier to keep a pawn in place where it can be attacked and then taken.
Preceding answers and comments make many good points.
Why struggle for a single pawn?
The simple answer is: Chess games are mostly won by material advantage!
Generally being up a single pawn in an otherwise safe position is a clear advantage, often a winning advantage.
In such a situation a GM analyst might say: And the rest is technique!
Of course there are myriad situations where the extra pawn does not guarantee a win but the general rule is that material advantage counts.
This is seen at the highest level eg Bobby Fischer was happy to take the so-called "poisoned pawn" variation in the Sicilian and hang on to it even though White had an initiative.
He knew that he if could hold on to that humble pawn it would lead to increasing advantage as pieces were exchanged.
The player with the extra pawn also has the option of giving it back at some stage in exchange for some other advantage.
As the game progresses and piece exchanges occur the value of that extra pawn increases.
The answer by @Brian Towers is both beautiful and true, but not always true. It is perhaps true if you think of every pawn as a potential queen. However, in endings, especially in Rook endings, the textbooks will tell you not to be drawn into passive defence but to keep your pieces active, and counterattack. That being said, you may not have the choice. There may be no counterattack available that is strong enough. And even if you do have a choice, it may not be obvious which is the right choice. You may not have a workable defence available either. It then becomes a matter of style, which of two evils to endure.
You should think of pawns not as individual units but as a group acting as one unit. If one pawn is one "point" then the entire pawn formation is 8 "points". This actually makes pawns one of the most powerful units in the game. The pawns are your front line and do an excellent job of defending each other in diagonal lines. If your opponent takes one of your pawns then your front line has been split into two smaller pawn formations. This is a weaker line of defense because now you have a hole open between the two formations. Your opponent could place a rook or queen on the newly opened file and be able to threaten pieces behind your front line. So you are defending more than "just a pawn", you are defending the pawn structure as a whole.