I've been reading some topics, and people are saying that if Dagoth Ur was to be successful in Morrowind, the Oblivion Crisis would have never happened.

Can someone explain to me how and why?

The return of the Heart of Lorkahn knocked Red Mountain out of commission as a tower, which made the barrier between Oblivion and Mundus less stable. I don't think that means that the Oblivion Crisis couldn't have happened, just that it was harder to make happen. http://imperial-library.info/content/nu-mantia-intercept-letter-3 has more.

Plus, if Dagoth Ur had been successful he would have eventually made war on Cyrodiil, and with Akulakhan he would have won easily. Even if the Mythic Dawn were still successful in summoning Dagon, Akulakhan would have smashed him to bits (that is if Dagoth Ur didn't dispose of him first). Dagon and his armies would have nothing on Dagoth Ur wielding a Numidium-on-steroids with ash armies to boot.

Mythic Dawn would likely be destroyed before then - I do not doubt that even Dagoth Ur had no love for the Daedric Princes of the Four Corners - especially Dagon.

Hmmmm, what would a Dagoth Ur ruled world be like

You've seen corpus monsters right? you shouldn't need a further vision of Dagoth Ur Nirn.

oh yeahhhh.....