In the first meaning of begin, it is possible to use the -ing form or the to + verb (infinitive) form after it:
She stood up and began playing or to play the trumpet.
With the -ing form we expect the trumpet playing to go on for a while, and assume that in the past (perhaps) there has been some playing.
How is it possible for there to be some playing in the past when it has just began ? Perhaps she began to play again after recovering from some injury?
Can't you say She began to play again?