No, let me get that. This is on me.

• You are in a restaurant and the bill arrives...
• 'Let me get that' = let me pay that. Let me means allow me. It means: 'Don't pay it - I want to pay it.'
• Get, in this context, means pay.
• This is on me. This is another expression meaning I'm going to pay this bill.
• You don't need to say both of them together, although it's fine if you do.

This sentence is an example of food, restaurants, colloquial expressions.