Because tension is there in horizontal direction , and it acts for a however small but finite amount of time , hence , since a=F/m , and for a small time it acts , hence it must increase velocity by a x t. where t is the time for which it acts and F here is tension and m is mass of the ball/stone which is rotating in circular path
Kindly see the edit too
And if only tension is there , then acc. to Work Kinetic Energy theorem , the Kinetic energy must not change , hence forth , speed must not change , but it is changing here according to newton's laws