To eat an insect, the spider bites it and injects some venom. That venom disables the insect, but it also starts digesting and liquifying the insect. Since insects have a hard exoskeleton, the spider winds up with something very similar to one of those drink boxes of fruit juice. Even better, the spider's fangs give it built in straws.
It takes some time to liquify the insect, so the spider will often wrap the insects in silk to keep them safe until they are ready to eat.