I understand your concern - it is a big responsibility. But children with allergies must be given the chance to attend parties and play dates like anyone else. As PP suggested, ask the child's mum to show you exactly how to use the Epipen. You will also need to know what symptoms the child is likely to display if he is having a reaction (eg itching, hives, swollen lips). Ask if he needs anything else apart from the Epipen, eg Piriton. AFAIK, if you use an Epipen, then you also need to call an ambulance because the person will still need specialist treatment (IV fluids etc), but paramedics can start these.
You will need to check if the child should avoid foods that say "may contain nuts", "produced in a factory that also handles nuts" etc, or just food where nuts are an ingredient. Check the labels of whatever you buy carefully. Remember also to check the accompaniments to your meal, so if you serve fish fingers, check the ketchup ingredients too!
You can minimise risk still further by ensuring the food preparation area and utensils are completely clean, reminding the child to wash his hands before eating, and wiping down the table and chair where he'll be eating very thoroughly beforehand. This helps to prevent accidental cross-contamination from nut traces that are around most people's homes. But he's probably not that sensitive, or else he would be having frequent allergic symptoms at school from fellow pupils who have peanut butter for breakfast and leave traces of it on door handles, books etc.
It sounds a lot but these extra preparations only take a few minutes and might help you feel more confident in hosting your DC's friend. And once you've done it once, you'll feel more relaxed about it in the future.