A flexible budget is a budget that changes depending on how much product you sell or manufacture.
Let’s say you pay your salesmen a 10% commission and you plan to sell $100,000. You would budget your commission expense at .10 x $100,000 or $10,000. But what if the salesmen sell $120,000 in product and you pay them .10 x $120,000 or $12,000? If you were doing a static or unchanging budget you would say that you were $2,000 over budget and berate your managers for going over budget. That would just encourage them to limit sales and limit profits. It makes more sense to have a budget that takes into account that increased sales will cause you to incur increased expenses.