Gripa means 'seize, grab'. Fatt originally means '(a) hold, grip', but is now used mostly in special phrases. Gripa fatt (i något) literally means 'grab a hold (of something)'.
In your example, it is metaphorical and means 'really work/focus/concentrate'.
I wouldn't say it is regional. I would rather say "ta tag i" as MattiasNYC suggests. with "fatt" I would have used "få fatt i" ie to catch somthing.
"Gripa fatt i" would - to me - be an older expression.