'...Then Black must stake a bigger claim in the center and advance either his c6- or e6-****, challenging White's central position and striking out for still more space.'
In this sense, soheil, strike out means to to hit or attack. "Because of his anger, he struck out with some very hurtful words which saddened his wife."
challenging White's central position and striking out for still more space.'-- I think 'strike out' here means 'set forth resolutely', like a pioneer or adventurer.