This is one of those words that covers many possibilities.
In it's plain form, it simply means to untangle something, straighten a thing out, whether it's a ball of string or a misunderstanding.
But alongside that, it has the sense of fending for yourself, of getting by.
Slip in the reflexive se and you have a phrase that is so useful.
You can manage, you can get by, you can sort it out.
You'll be OK, you won't need a helping hand, you'll muddle through.
You'll deal with it.
Tottering out of the supermarket, two tins of anchovies and a tub of olives in my coat pockets, a bag of frozen peas under my hat, two bottles of wine under my arms and a pile of boxes balanced under my chin. I forgot my bag again (absolutely no giveaway bags in French supermarchés), but je me débrouille.
I spent the week in bed with flu, there's nothing in the fridge and the cat is glaring at me. My friend calls and asks if I need anything, but guess what? Je me débrouille.
A plumbing emergency has me dashing to M Bricolage but they're fresh out of the widget I need. Never mind, je me débrouillerais with an empty yoghurt carton and some black tape.
Je me debrouillerai. It's my blog, I'll do it my way.