Translation:The young woman lies down on the bed beside the man.
the hint under befekszik is "lay into" or "is laying into". To lay requires a direct object in English, unlike to lie. Here the woman lies in the bed is correct. My question: isn't fekszik an intransitive verb? in that case, it can never be used in the definite form and can not be translated by "lay".
This clashes a bit. Befekszik means to "lie down into something", and az ágyra is "onto the bed". A possible situation where this would apply is if the bed were inside a box, but that's rather unconventional.
Either use "lefekszik az ágyra" - lie down on(to) the bed, or "befekszik az ágyba" - lie down in(to) the bed.
Although the former option is a possible interpretation if you try hard enough, this sentence is mostly understood as the man is already in bed and the woman joins him.
The meaning of bed-is-next-to-man can be better expressed simply by "az ágyba a férfi mellett". Or "a férfi melletti ágyba". Or "az ágyba, amely a férfi mellett van".