How does austerity (reducing government expenditures in order to control/safeguard public-sector debt) cause the end of a recession (shift from negative to positive GDP growth rate)?

Saying austerity led to the end of a recession is like saying that cutting costs at your restaurant led to an increase in customers, or allowed you to charge higher prices for your food.

Greece and Italy are fighting austerity because they take the traditional economic stance - government needs to spend tax dollars on increasing investment, education, employment, etc. instead of cutting investment costs and ensuring tax money is channeled to interest payments.

Perhaps foreign investors saw that Portugal was taking its austerity seriously and began to increase investment, reassured their money would be relatively safe. But how could austerity lead to the boom in tourism and exports cited in the NY Times article?