Companies, and the people that run them, own a lot of that stock. By buying some, you're increasing the demand, and therefore the price, and therefore the amount of value owned by the company and its leadership.
The act of buying stock signals demand and reduces supply, which will effectively cause the price to rise. Indirectly, rising stock prices are a positive indicator for the health of the company. Directly, if the company chooses to sell stock, it will be able to sell it at the higher price.