Adding salt to water when boiling eggs doesn't significantly affect the cooking process, but it can help prevent the egg from cracking and may make it slightly easier to peel.
The primary reason people add salt to water when boiling eggs is to increase the water's density. A denser solution can reduce the likelihood of the egg cracking during boiling due to the osmotic pressure difference between the egg's contents and the surrounding water. If the egg does crack, the salt can help coagulate the egg white more quickly, potentially minimizing the amount that leaks out. While the effect is subtle, it can be beneficial, especially if you're boiling a large batch of eggs or if some eggs have hairline cracks. Some also believe that salted water penetrates the shell slightly, making the egg easier to peel after cooking, although this is debated.
For easier peeling, try shocking the cooked eggs in ice water immediately after boiling. The rapid temperature change causes the egg to contract slightly, separating it from the shell membrane.