Foreign keys enforce referential integrity. These constraints guarantee that a row in a table order_details with a field order_id referencing an orders table will never have an order_id value that doesn't exist in the orders table.
Foreign keys aren't required to have a working relational database (in fact MySQL's default storage engine doesn't support FKs), but they are definitely essential to avoid broken relationships and orphan rows (ie. referential integrity). The ability to enforce referential integrity at the database level is required for the C in ACID to stand.
As for your concerns regarding performance, in general there's a performance cost, but will probably be negligible. I suggest putting in all your foreign key constraints, and only experiment without them if you have real performance issues that you cannot solve otherwise.
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