There's no clear canon answer, but from all my reading about the game, I'll try and explain how I think it most probably works as best I can. To do so though, I'll have to go into how Tenno work a little bit, or at least my educated guess as to that.
Tenno were once what we would call human. Flesh and blood, cells and DNA. Within them, like all humans, they bore what we might call a soul, a structure of possibly-transdimensional energy patterns that encoded their consciousness. The Orokin used the technology we know as Oro to store these "souls" and move them from one body to another through Continuity, using Kuva to "prime" the host body for consciousness implantation. However, when the Zariman 10-0 experienced a catastrophic Solar Rail failure and was cast adrift in the Void, this had an unexpected side effect on the to-be Tenno. For reasons unknown, their consciousness, their soul, was somehow bonded with Void energy, fundamentally altering the nature of their existence. It was unclear at first what exactly had changed, but it seems the Tenno gained the ability to connect with the minds of other beings, even subsentient robots, and even merge with them in some cases. They also seem to have gained the ability to focus the Void energy bound within them into destructive forces wielded as a weapon, or to shift themselves slightly into the Void and go unseen. However, it's not until The War Within that we get the first clear idea of what the Tenno truly are now, and not until the Sacrifice and other subsequent quests that we learn much further. To cut a long and convoluted story short, the Tenno appear to be part-energy beings, still resident in the plane of existence we know as the material universe, but no longer entirely bound to it. Tenno have physical bodies that can be injured and killed, but it seems like said bodies are more of a material projection than anything, given that A: During and after the last segment of The War Within, we can shift in and out of our Warframes at will, most clearly illustrated in the cutscene on the Kuva Fortress, where our Operator materializes out of our Warframe's chest, after having literally teleported themselves there across hundreds of thousands of kilometers of empty space, from the Somatic Relay in the Orbiter. (it's worth noting as an aside here that, from studying the Tenno, the Orokin found a way to crudely replicate Tenno transference, and created the Somatic Relay from it. This device enabled any human who sat in it to use Transference, but only to a limited, clumsy extent, unlike the ease with which the Tenno could merge and unmerge with the souls of other beings.) B: During the War Within and the Sacrifice's Operator-only segments, dying will briefly show your Operator floating in the Void in a fetal position, before respawning. Now this may sound like grasping at straws here, given that it's ultimately a game mechanic, but I believe, given everything else we see and hear about Tenno, that this is actually a visualization of what would happen if a Tenno's physical form was actually killed - being part-energy beings, a fully-awakened Tenno would reincarnate into a new body after being "killed".
Now, as for Warframes, that's perhaps a bit simpler. Warframes are technically Human/Technocyte hybrids. (Technocyte being the name of the bio-mechanical blight that created the Infested; the Helminth in our Orbiter is another strain of Technocyte.) The Orokin originally made them independent of the Tenno, as biomechanical warriors to fight the Sentients. However, no matter how much the Orokin researchers drugged, tinkered, and tormented the prototypes, they could not control them - the prototype Warframes, overcome with unfathomable pain, anger, and confusion, simply lashed out at everything in sight. However, the Tenno, with their unique ability to connect to and merge with other beings, were able to join their mind to a Warframe's, and, as our old "friend" Ballas put it, take away their pain. How do Tenno actually control the Warframe? In short, when a Tenno transfers into a Warframe, their souls are essentially combined temporarily, essentially creating a new sort of being, more Tenno than Warframe in mind, but still unique. Basically, Transference allows a Tenno to become the Warframe, which, as Nutty mentioned above, explains how everything up until The Second Dream happened - the Tenno were in a deep, artificially-induced sleep in a suspension pod on the Moon, and effectively controlling their Warframe in a sort of lucid dream.
Hope this answers some of your questions.