As Frank pointed out, you need to use DISTINCT. Also, since you are using composite primary keys (which is perfectly fine, BTW) you need to make sure that you use the whole key in your joins:
SELECT
P.pe_name,
COUNT(DISTINCT O.ord_id) AS num_orders,
COUNT(I.item_id) AS num_items
FROM
People P
INNER JOIN Orders O ON
O.pe_id = P.pe_id
INNER JOIN Items I ON
I.ord_id = O.ord_id AND
I.pe_id = O.pe_id
GROUP BY
P.pe_name
Without I.ord_id = O.ord_id it was joining each item row to every order row for a person.