The Short Answer:
The gravity field used by a tractor beam simply overwhelmed the relatively fragile fighter aircraft of the 20th century and crushed them, due to their lack of internal structural integrity fields and inferior construction as compared to superior armor and construction processes used in the design of the 23rd century starships.
A Longer Answer:
In Star Trek, where the tractor beam gained its cult popularity, it was a field of force designed to hold, tow, or control smaller vessels using an "attenuated graviton beam." The very utilitarian nature of the tractor beam, belies its potential destructive power. Indeed, the Borg use their tractor beams as weapons to hold ships in place, drain their shields and increase their targeting ability on a non-moving ship.
This field could reinforce structural integrity in vessels from the same era because it was designed to interact with the molecular structure of those ships. However, when used against ships or structures of a different era, particularly those without the benefit of structural integrity reinforcement, the tractor beam was the equivalent of a gravity weapon, tearing the vehicles apart.
While tractor beams were commonly used in the Federation and among Alpha Quadrant species, the process was not always a perfect one as different races found their tractor beams having different effects on each other's ships. A Romulan tractor beam was discovered to have a shearing effect (causing microscopic deformations in the hulls of Federation ships. (TNG: The Mind's Eye)
It is this same shearing effect, which tore apart the relatively flimsy 20th century fighter aircraft when a tractor beam was used to attempt to push them away from the Enterprise as it descended into low Earth orbit. The metallic hulls of those planes were simply not designed to withstand the force of the tractor and could not provide sufficient resistance to the equivalent of a giant fist squeezing the plane from the outside.
It is easy to forget how durable most Federation or 23rd century technology is in comparison to our own because their weaponry is so powerful. Most of the weapons of modern Earth could barely scratch an unshielded Federation starship. The molecular processes used to design their hulls make them incredibly strong and they are able to be reinforced with structural integrity fields, which constantly increase the rigidity and resistance to stresses from motion, inertia and other movement stresses up to and including moving near the speed of light! Modern Earth aircraft only have the strength of their materials, construction and design to help them withstand their operating stresses which a tractor beam can easily exceed.
As far as using the tractor beam on people, I cannot recall an instance where a shipboard tractor was ever used on an unprotected human, and if it was, it was probably highly modified in order not to harm the human. Normal ship tractors would and should crush unprotected human beings unless highly calibrated.
The episode in question is: "Tomorrow Is Yesterday" a first-season episode of Star Trek: The Original Series. It is episode #19, production #21, first broadcast on January 26, 1967, repeated July 13, 1967, and was remastered in 2006 for syndication broadcast on May 5, 2007. The teleplay was written by D.C. Fontana and directed by Michael O'Herlihy.