LOS ANGELES—Walt Disney Co. rose to the top of Hollywood on the strength of its ability to pack movie theaters and theme parks. Now it is betting that to stay on top, it needs to come to you—by streaming entertainment directly to TVs and phones.

Disney is preparing to launch a service called Disney+ that will, for a monthly fee, stream new and old shows built around its most popular franchises. That means breaking off a lucrative relationship with Netflix Inc. and instead competing against the streaming powerhouse. On Thursday,...