big leagues, throw the pass, and run the race?

You mean apart from being really bad writing? The author is mixing a series of sports metaphors. To step into the big leagues means to join in the competition at the highest levels. To throw the pass means to take risks--in American football, throwing a pass is risky, but the rewards can be great. To run the race means that one goes out into the competitive marketplace and puts out the effort to succeed. That last figure of speech, to run the race, leads back to the theme. One has to get involved. One cannot wait for opportunities to come along--one cannot procrastinate, one cannot put off making the effort necessary to succeed, one has to get involved, now, one has to run the race.