How do exchanges keep relatively similar price on coins?
If I could buy a bitcoin on Exchange A for $2,000, turn around and immediately sell it on Exchange B for $2,500, and make an immediate profit of $500, I would do that all day long, and get rich!

But if even a small price discrepancy exists between two exchanges, many people will find it, and start buying on one/selling on the other. 

All that one-sided buying in one market and selling in the other market moves the price back to equilibrium.