They have distinct courts set up for business-related cases. In most states, if you file a lawsuit, you may not end up in front of a judge that specializes in business law and litigation, which may or may not work to your advantage. In Delaware, you will. Every corporate law firm also has experience with Delaware code (many have dedicated DE code experts). Law firms have standard document templates for Delaware. Every VC has experience investing in Delaware corporations. Courts are highly funded in Delaware relative to most other states, so cases can proceed more rapidly. Even the court clerks are dedicated to corporate law.
There is very little incentive for start-ups to incorporate elsewhere.
Below it is suggested that Delaware is cheaper, in fact it can be much more expensive for this reason.
edit (re grandparent): Delaware is definitely not a "safe haven" for litigation. It's more predictable for the basics, but it isn't going to prevent you from an Eastern Texas District Court summons.