I feel like there must be some equivalent to the phrase "looking under the hood"in British English but have drawn a blank. Can anybody help?
I assume you are referring to what Americans call the hood of a car.
In Britain and in various other countries where English is spoken it is always called the bonnet. And we look under the bonnet for exactly the same reasons that Americans look under the hood.
I haven't heard much use of under the bonnet for the metaphorical use you ask about in your comment. For this it's more common in the UK to say something like get your hands dirty/mucky (from farming/gardening).
This can mean to involve yourself in all parts of a job, including the parts that are unpleasant, or involve hard, practical work. It can also be used to mean 'get stuck in there and see what's really going on here' -
Unlike other bosses, she's not afraid to get her hands dirty, and the staff like that.
I am familiar with the usage, 'Take a look inside the box', but I can't ascribe it to any particular version of English. There is a variant on this, 'Think inside the box', which at first doesn't sound as clever or valuable as the more familiar, 'Thinking outside the box'. But of course the point is that if you can't achieve or analyse anything in conventional terms ('Inside the box') you possibly aren't functioning particularly well, or well positioned to make useful observations 'Outside the box'.
An alternative, and perhaps a more British one, is, 'To see what makes it tick', or 'Let's see what makes it tick'. This brings to mind an earlier question: What does the phrase 'make the girls tick' mean?.