Although illustrations in medieval manuscripts often take a creative approach to reality, they can give a good interpretation of how the ladders were used during an escalade.
There are depictions of ladders being held against on the walls in various ways, such as with armored hooks at the top or by bracing it at the bottom, and never seem to depict ladders being pushed or falling away from the walls. Soldiers could confidently survive the climb because of sheer numbers: a castle would have significantly fewer defenders than the army sieging it, leaving relatively few defenders per section of wall being attacked.
In several manuscripts, soldiers on the ground can be seen holding onto the ladders from underneath:
In the bottom middle of first image, one soldier is underneath the ladder and facing away from the actual fight, and appears to just be bracing the ladder while other soldiers climb up. In the bottom right of the second image you can see a soldier holding a shield in one hand and a rung of the ladder in the other, and unlike the other soldiers he's depicted standing behind the ladder rather than trying to climb it. Pushing an empty ladder away from the wall or to the side is easy, pushing a ladder with several armored soldiers on it and several others bracing it certainly isn't.
Additionally, the top of the ladder could be hooked, and if the hooks reached up and over the parapet then the ladder would need to be lifted some amount before it could be pushed away from the wall. Ropes or poles could be attached to such a ladder to help soldiers add additional weight and leverage:
The first image simply shows hooked ladders leaning against walls, however the second and third demonstrate how they could be held in place with extra support. The second shows two solders holding ropes attached to the armored top of a ladder, and the third image depicts some sort of flexible ladder held in place by poles.
It appears that ladders could also be secured in place by pegs at the base:
The first image shows the end of a rope or chain ladder being staked into the ground, to prevent the defenders from simply pulling it up and over the wall. In the second image a solder in the bottom left is hammering a wedge into the base of the ladder (though I can't imagine how that particular wedge would help, securing the foundation of the ladder could certainly make it harder to push).
Finally, there's the question about how soldiers survived the climb to the top of the wall. Although the illustrations leave out a lot of details one thing is consistent in all of the above images: there are more soldiers attacking the walls than defending the walls.
During a siege every soldier was another mouth to feed, so garrisons were as small as they could be while still manning the castle defenses effectively. As an example of the force multiplication a castle can provide, in 1643 while residing at Corfe Castle in Southern England Lady Mary Bankes grew the garrison from an initial 5 defenders to 80, and they were able to hold off a sieging force of 500-600 with only a handful of casualties. In 1403, the entirety of Caernarfon Castle in Wales was successfully defended by a garrison of just 30-40.
So, the soldiers climbing these ladders wouldn't just be cut down one-by-one because the section of wall they were assaulting likely wouldn't have many defenders. Additionally, attacks would be supported by archers and assaults on other parts of the castle, making any individual ladder a survivable climb if you had numbers on your side:
All of the images above come from a site called Manuscript Miniatures. As they say here they offer only images from manuscripts that have been in the public domain for centuries, so hopefully there are no copyright issues.
The manuscripts these images came from and links to the original images, in order of appearance:
BL Yates Thompson 12 Histoire d'Outremer, image here
BL Royal 20 C IV De Casibus Virorum Illustrium, image here
BL Royal 10 E IV Decretals of Gregory IX, image here
Besan?on BM MS.1360 Bellifortis of Konrad Kyeser, image here
ONB Cod. 3062 Kreigsbuch, image here and here
Christ Church MS 92 De Nobilitatibus, Sapientiis, et Prudentiis Regum, image here
