Why do companies buy back their shares?

Financial engineering to create wealth effect

Because if they distribute it via dividends they are exposed to another layer of tax. Withholding tax for foreign holders and personal tax for local owners.

You will be surprised to know that a significant percentage of growth of earnings come from stock repurchases.

Although note that buying back stock no matter what is not a good nor viable strategy. Strategic buybacks below valuation shows good competence of management.