arreglar vs arreglarse
Arreglar is fixing, tidying or sorting something. Arreglarse is getting ready and managing — the reflexive brings it back to you.
arreglar
To repair, tidy up, sort out or alter.
Arregló el coche en un día.
He fixed the car in a day.
Repairing. 'Reparar' also works but is more formal.
Arregla tu habitación.
Tidy up your room.
Tidying. Also 'ordenar'.
Ya lo arreglaré con ella.
I'll sort it out with her.
Resolving a problem between people.
arreglarse
To get ready, dress up, manage, or work out.
Me arreglo para salir.
I'm getting ready to go out.
Dressing up, doing hair and makeup.
Ya me las arreglaré.
I'll manage.
Arreglárselas = to cope, to get by. Fixed phrase.
Todo se arreglará.
Everything will sort itself out.
Se impersonal: things work out.
Tricky bit
'Arreglarse el pelo' (do your hair) vs 'arreglar el pelo a alguien' (do someone's hair). The reflexive tells you whose hair it is.