It is very helpful to have played on rope swings and chain swings as a child. That trains the intuition well. One can most definitely feel the increase in rope tension at the bottom of the swing's path and the easing of tension at the ends.
There are two effects going on. [Or at least one can account for things that way]. Both are most obvious when one is swinging very high. Both work to increase rope tension at the bottom of the swing path and decrease it at the ends.
The first effect has to do with components. The rope tension acts radially. Gravity acts vertically. The rope only has to resists that component of gravity which acts in a radial direction. If the swing is in a very high path so that the rope gets horizontal, the rope goes completely slack -- the component of gravity acting in the radial direction is zero. At the bottom of the path, gravity acts exactly in the radial direction and its contribution to rope tension is at a maximum.
The second effect has to do with centrifugal force (or with centripetal acceleration as your physics teacher may prefer to have you think). At the bottom of the swing's path when the rider is going fastest, the centripetal acceleration is the greatest. The rope tension supplies this centripetal acceleration.
When "pumping" very high swings (think circus trapeze rather than playground swings) at high amplitudes, the proper approach is to lift one's body at the bottom of the stroke and extend it back out at the ends -- doing work and injecting energy into the system when the resting force is greatest. If you can pump up to near horizontal, you'll be facing about 2 g's of tension at the bottom of the stroke and 0 g's at the ends (remembered, not calculated).
It is not easy to deal with 2 g's with hand strength (rope swings). I had the great good fortune to also have access to a 15 foot chain swing with a wooden "stand-up" bench. Getting that thing up to horizontal was not difficult. Getting it much past horizontal was very challenging. You lose a lot of energy and significant control when the chains go slack and then snap back into tension. I suspect that skate boarders face much the same challenge for identical reasons, feathering out their foot pressure above the top of a half-pipe.