In holographic duality, the two mathematical theories (e.g. a string theory in AdS space, and a conformal field theory in flat space of one less dimension) are believed to be alternative descriptions of exactly the same thing.
You might say, but in the most famous example, the AdS space has ten dimensions, and the flat space has four dimensions, how can they be the same thing? The answer is that the boundary-at-infinity of the AdS space only has four dimensions. The activity of the "fields in the flat space" describe the entry and exit of strings from the boundary-at-infinity, into the "center" (the bulk) of the AdS space.
From the perspective of the four-dimensional theory, the radial direction of the AdS space corresponds to the size of objects in the field theory. The larger an object is in the four dimensions, the deeper it is in a fifth dimension of AdS space. So the four-dimensional view is (metaphorically) like many slices of five dimensions, that have then been superimposed on each other.
So that's five out of the ten dimensions. The other five dimensions are closed on themselves, and correspond to field strengths in the four dimensions, in a way that I am a little unsure about.
The actual way that four dimensions corresponds to ten dimensions is still a little mysterious - not all the details are worked out. The duality itself was guessed by Maldacena (in this paper) after he studied four-dimensional black holes in ten-dimensional string theory. It seemed like the inside of the black hole should be described in two ways, one four-dimensional, the other ten-dimensional. Mathematical equivalence is not absolutely proven, but it has passed every test (when you calculate two things in the different frameworks, that should be equal, they come out equal).