I can't answer your question in general, but I can explain the main ideas of the game. Here is the game (actually the game lasted a few more moves, but at the end position here it is lost for Short):
1. e4 e6 2. d4 d5 3. Nd2 c5 4. Ngf3 Nf6 5. exd5 exd5 6. Bb5+ Bd7 7. Bxd7+ Nbxd7 8. O-O Be7 9. Re1 O-O 10. Nf1 Re8 11. dxc5 Nxc5 12. Be3 a6 13. c3 Rc8 14. Qc2 Qc7 15. Rad1 b5 16. Bd4 Nce4 17. Ne3 Rcd8 18. a3 Bf8 19. Qb3 Qc6 20. Ne5 Qc7 21. Nd3 Bd6 22. g3 Qc8 23. Bxf6 Nxf6 24. Nb4 Bxb4 25. axb4 h6 26. Rd4 Re5 27.Red1 Qh3 28. Qc2 Rde8 29. Rh4 Qe6 30. Rhd4 Qh3 31. Rh4 Qe6 32. Qd3 Kf8 33. Rd4 Qh3 34. Qf1 Qe6 35. R1d3 Re4 36. Qd1 Rxd4 37. Rxd4 Rd8 38. Kg2 Rd7 39. Qd3 g6 40. h3 Kg7 41. Qd1 h5 42. Qf3 Ne4 43. Nxd5 Ng5 44. Nf4 Qb6 45. Qd3 Qc6+ 46. f3 Rxd4 47. Qxd4+ Kh7 48. Qd5 Qf6 49. h4 Ne6 50. Nxe6 fxe6 51. Qb7+ Kh6 52. Qxa6 Qe5 53. Qb6 Qe2+ 54. Qf2 Qd1 55. Qe3+ Kg7 56. Kf2 Qh1 57. Qxe6 Qh2+ 58. Ke3 Qxg3 59. Qe7+ Kg8 60. Qg5 Qe1+ 61. Kd4 Kf7 62. Kc5 Qf2+ 63. Kxb5 Qxf3 64. c4
Take a look at the position after Black's 16th move. The opening phase is over. It is a typical isolated queen's pawn position. Here Kasparov has a better pawn structure as his pawns are in two pawn islands as opposed to Short's three pawn islands. Why does that matter? A pawn island is pawns that can defend each other. As you will see the only pawn on the board that can't be protected by another pawn (except after an exchange) is the d5 pawn.
As compensation for this tiny disadvantage black normally gets more active pieces. As you can see Short has good pieces here. If Kasparov can manage to get Short on the defence then this compensation (active pieces) will disappear. This might not be much, but at the top level any kind of advantage is better than none.
Why can't Kasparov attack Short's king instead, or some other piece? Well, any piece can always move (but remember that the d5 pawn cannot move) and since Short has more active pieces any direct attack will fail.
What did Kasparov gain from attacking the pawn here? Well, he forced Short to give up his bishop against a knight. A bishop is considered slightly stronger than a knight, in particular in positions with pawns on both wings.
Still the d5 pawn is the easiest target to attack, although now the a6 pawn has become weak as well. It would be dangerous to attack this pawn though, as doubling rooks on the a file would leave the white king less well protected. This is something Short would have loved as it could have given him a strong counter attack.
During this phase Short tactically defended the d5 pawn by putting his queen on h3. Should Kasparov take the d5 pawn it would have let to a mating attack for black after rook to e1.
Still the d5 pawn is the only real weakness in Short's position. It still cannot move and should it drop the endgame is very difficult for Short. The plan for White would then simply to be to exchange all the pieces and win the pawn endgame. That would not necessarily be easy, but would give Kasparov an easy plan.
Moves 42 and out
Finally, on move 42 Short blunders tactically and loses the d5 pawn. Now the game changes character. Kasparov tried his best to exchange pieces while Short tries his best to avoid trading pieces. Short's best hope here is to trade pawns, hoping for a draw. Notice how Short avoids trading queens.
I only found the score up to move 64. Short played on, but finally he surrendered when he realised that Kasparov would get another queen by promoting the b pawn.
The pressure on the d5 square led to Short finally blundering the pawn. Had he not the game might have ended in a draw. Of course, there are other plans, but you have to pick one. Kasparov looked for the point in Short's position where he could attack and found it on d5 as the pawn could hardly move without being lost and could not be protected by another pawn.
Really strong players will find small ideas like that. Kasparov did not necessarily expect to win the pawn, but at least to force Short on the defence. If everything else is equal, having the more aggressive position is an advantage. Although it could be small, it is better than nothing.