Netflix is approaching - or already larger than - the size of Disney's comparable entertainment business, and Netflix is growing relatively fast whereas Disney's business isn't growing much or at all.
Disney's studio division does $8.3 billion in sales. Their media networks arm, which includes ABC & ESPN, does $23 billion.
Netflix will hit ~$14-$15 billion in sales this year. Apples to apples in terms of where they compete, Netflix will be comparable to or larger than Disney in size in 2018 or 2019.
Five years from now Disney isn't likely to be much larger than they are today (just look at their growth the last few years, flat to minimum). Netflix will very likely be over $25 billion in sales at that point (which would plausibly be larger than Disney's largest business, the media networks group).
At the scale Netflix has reached, every dollar of new growth that Netflix is obtaining is taking some bit of revenue away from Disney. Disney is going to mostly cannabilize themselves in switching to their own streaming service, including the mess at ESPN (the end of the hyper lucrative, subsidized cable & satellite subscribers that have been artificially fattening Disney's entertainment wallet for years).
Disney is mostly in a scenario of attempting to hold their ground. The same place Walmart is in with Amazon and for the exact same reason (minimum overall market expansion, low consumer spending growth across the developed world, while the new competitor is starting to eat your existing house).
Interestingly another similarity between those situations: Amazon is willing to compete at near zero profit generation in retail to pursue sales growth, which is brutal to an established giant like Walmart who has a huge investor base that expects them to generate considerable profits every year. Walmart can't just drop their profits dramatically to compete, their stock would implode. Disney is facing the same storm: Netflix operates at a level of minimal profit (~4%-5% net income margins), whereas Disney's long-established shareholders expect rich profits and a dividend (Disney typically sees 10%-20% margins in their entertainment groups, with a 16% overall net income margin). Netflix is willing to plow almost every dollar back into content production and licensing, and their shareholders cheer that on so long as the top-line growth continues.