The essential meaning of 'climb' is to ascend, to move upwards, go up a slope, incline, or staircase.
With a truck, the driver's cabin is higher off the ground than with a car, hence, you have to 'climb into' a truck.
For a small child to get out of a cot, he has to climb over the high railings.
'clamber' combines the meaning of 'climb' with the idea that the climb is awkward, difficult and laborious such that the person typically has to use both hands to hold on to what's available to help drag them up and so assist them in making the ascent, whether up a steep embankment, or out of a trench.
So - why do we say 'climb into a truck' and 'climb out of a cot', since we would also use our hands to hold on to something and drag ourselves up?
With 'clamber', we emphasize that the climbing, or climbing out of something, is difficult and awkward -(we can have a pleasant time, climbing/walking up a hill).
In 'climbing', the body is essential vertical, upright (depending on how steep the slope is)
When we clamber: firstly, imagine walking/crawling on the floor, using your hands, but with your feet still on the floor, not your knees as in crawling. Then imagine doing this up a steep slope, where the hands are holding on to jutting rocks, or vegetation to helps us, and the feet finding footholds where they can. The trunk of our body may be fairly upright as such (because we are ascending), but close to but not touching what we are climbing because we need to be close, to grab hold of what is available to help lift ourselves up.