Please excuse my ignorance - why do they have both a commercial license and an MIT license? I'm confused. If I want to use it on my commercial site, don't I have to pay for a commercial license? If so, why should I care whether they have an MIT license also? Sorry, licensing is something I never really looked into.
Dual licensing is not mysterious. I'm not sure why folks find it so hard to imagine that a product can be had under multiple terms, depending on your preference and requirements.
The MIT license is a couple steps up from public domain (and far more liberal than the GPL). Ext Core can be used under those terms. ExtJS (not the same as Ext Core) has different terms (I think it's a modified GPL and a commercial license).
You do not need a commercial license to use Ext Core in any circumstance. You can use it on a commercial site, you can distribute it in commercial (or Open Source) products, and you can make modified versions without returning changes to the maintainers. The MIT license does have some clauses about IP (like trademarks) and indemnity (i.e. they provide none).