Stallone's expression is a out-dated one. The context of the statement is that Stallone usually and apparently does most of his on stunts. He is asked about this in the interview. He answers that he does do most of his own stunts in a rather roundabout way. The "race horse" expression is here used. The idiomatic metaphor which Stallone cites is the example of the race horse who is old and thinks "just get me up to the gate and let someone else run."
Stallone uses the metaphor to convey how this is not true of him. Stallone is trying to explain how he is willing and able to perform any stunts that are asked of him and how he is not like the "old racehorse." It's an interesting way of expressing his feelings. It is Stallone's way of expressing how he is eager to still participate in his own stunts.