Why bemásznak and not only másznak? Bemásznak (crawling, climbing IN) fits with crawling into the bed, but not, when the spiders are crawling under the bed. They are crawling IN UNDER the bed?
It goes along with the postposition alá - that means they're going there (not just that they're under the bed already, crawling around). You need a "going to someplace" kind of verb. The area under the bed is seen as an enclosed space with an "inside", so bemászik works.