Translation:The small green bug crawls through the window into the room.
I guess "be-" is referring more to the room here ("into the room") than to the window ("through the window"). The focus is on the room.
But "átmászik" is not necessarily wrong, but a bit unusual to use with a window. Somebody either "bemászik" or "kimászik" "az ablakon", not "átmászik". The target is more important. One thing where "átmászik" is more appropriate is the fence: "átmászik a kerítésen". But you can also say "kimászik" and "bemászik" there, if the direction is more important.
If I really wanted to use "átmászik" here, I would say:
"... átmászik az ablakon, be a szobába." - through the window, into the room. You can think of the above as the simplification of this structure, as "through the window" does not really give us more information. So, it is much better to say:
"... bemászik az ablakon a szobába."
Good. I also tried to link bemaszik to szobába, which gives me climbing into the windows. This lives az ablakon which I believe is on the window. I was therefore looking for the "through", somewhere. Obviously, if the poor bug could get into the room, it had to go through the window, in this case.
You mean "into the room", not "into the windows", in your first sentence, right?
Yes, basically that is the logic. And if you really want to find the "through" part, well, it is "ablakON", which is not necessarily "through" but maybe the remnants of it.
The full version would be "az ablakon át" or "az ablakon keresztül". But the "át"/"keresztül" part is really optional here, there is more than enough movement indicators in the sentence without them.
But if you want to say it, you can say
"... bemászik az ablakon keresztül a szobába."
And then we have everything covered. Right?
I think vvsey has explained my difficulty with this. It's so imprecise and that's one of the biggest problems I have with Hungarian. In English, the geography of what happens is fully mapped out with through and into. In Hungarian we know it goes into the room but with the window maybe the transporter room of the Starship Enterprise had a part to play because we aren't told.