As the match goes on, I notice the respawn time increases significantly. Why is that? Is it number of deaths, length of the match, character level, or something else entirely?
The respawn time is determined solely by the hero level, following a very simple formula:
There are a few things which can lower that value. Bloodstone decreases your respawn time by 4 per charge. It is possible to spawn instantly if the number of charges in the Bloodstone is greater than the hero level. Techies' skill Suicide Squad, Attack! halves his spawn time when utilized.
You can also buyback to spawn instantly. However it is costly (100 + (Level*Level*1.5) + (GameTime*15)) and has a large cooldown. (240 seconds)
As the dota 2 wiki doesn't say anything conclusive on this matter, we can safely presume it follows the dota 1 mechanics.
So,the only thing that influences your dead-time is character level. Also, if you have bloodstone (based on the bloodstone charges you have) this reduces your time spent dead. Also, if you use Suicide from techies(Techies is not yet implemented) this will drastically reduce your time spent dead.