Decisions are engaging
This is true on every level: encounters (both combat and other wise), dungeons (or other local-level areas), campaigns, and adventures. Descriptions of settings and characters and the environment contribute to immersion and bringing the game to life, yes, but to be engaged with the gameplay your players should be making decisions.
Rolling to hit is not a decision
Player character Anne is adjacent to her opponent Ben. She rolls to hit him. He rolls to hit her. Eventually the dice kill or KO one of them. Neither makes a single decision, and Anne's player does not feel engaged.
Granted, this is a sterile environment - there will inevitably have been decisions leading to that fight. But the fight itself is just roll-to-kill.
A change of circumstances creates decision points
Anne and Ben fight. As Anne nears victory, Ben turns and flees. Does Anne give chase to finish, or is his retreat a satisfactory win?
Say Anne chases Ben. She catches up to him only to find he's convinced some guards that Anne is a murderous thief. Does Anne still want to fight Ben so badly, or does she turn herself in to the city guard, or does she now flee? Decision point.
Say Anne does not give chase. Later, Ben returns with reinforcements. Anne's earlier decision has real consequences down the line. Bringing me too...
Interesting decisions with consequences are engaging
If a fork in the road takes you down two identical paths and rejoin at your goal, you don't really have a worthwhile decision point. This is why sometimes, in combat, having an array of attacks or spells or manoeuvres does nothing to increase the fun of combat - they either all have the same end result, or one is so strictly superior that there'd be no reason to choose it. You resort to 'roll-to-kill'.
Engagement with decisions is driven by goals
Combat is not an encounter. It is a way of resolving an encounter. An encounter is where two parties have differing goals that create conflict with one another. The inevitability of combat is variable. As you seem to mostly be talking about combat, though, I will focus on that.
Your players will decide how to fight based on their goals. The way you describe things will affect their perception of such goals - do they wish to kill all in their way, or maximise their own survival rate, or achieve something else like retriving the Unholy MacGuffin?
Likewise, the characters and monsters they fight should have goals. Not everything will fight PCs till its last breath. And when backed into a corner, an NPC or monster might well change strategies drastically. The change of strategy can create decision points for the PCs - "...oh. Didn't know it could do that. Better fall back."
Play many videogames and this will be a familiar idea: a big baddy that, just when you think you have it beat, comes back with twice the ferocity. The more it changes about itself and the battlefield (within reason; it should be the same thing) the more it will rejuvenate any staleness the combat has gathered.
What you effectively do is turn one long combat encounter into a series of short ones. Each of big bad's 'forms' is less powerful than if you had created a one-phase boss. But the transformation should attempt to push the players to make new decisions.
I highly recommend the writings of The Angry GM on encounter design (including extensive discussion of decision points, the 'dramatic question', and conflict) and, latterly, boss monster design. Such articles are generally the source for what I have outlined above.