A higher voltage can be produced at 1:1 ratio because the transformer is being run in flyback mode. Unlike normal forward mode, the primary and secondary don't conduct at the same time.
The input pulse builds up current in the primary. This does produce the same voltage on the secondary. However, the secondary is wired so that is a negative voltage, which is blocked by the series diode (D1) in the schematic you show. As a result no current flows in the secondary while voltage is being applied to the primary.
Since no secondary current flows, the primary looks electrically like just an inductor. The energy going into the primary goes into building up the magnetic field in the transformer core, just like in a simple inductor.
The primary current is then abruptly shut off. The circuit is designed so that the only place the energy in the core can go is out the secondary. The primary being shut off causes a secondary pulse of opposite polarity as when the primary was being charged. This time D1 allows conduction.
During the output pulse, the current in the primary is 0, so the transformer is again acting like an ordinary inductor. This time it's the secondary that is conducting. The output pulse is just an inductor discharge. The current decreases proportional to the reverse voltage applied to the secondary, which is the output voltage (plus the drop across D1).
The interesting thing about flyback mode is that the the output/input voltage ratio is independent of the transformer ratio. The transformer ratio does dictate the reverse voltage each side sees when the other is conducting. If you use a 1:1 transformer to step up a voltage by 5x, then the primary sees a reverse voltage of 5x during the discharge pulse, compared to the voltage applied to the primary during the charge pulse. You have to consider these voltage stresses carefully, and trade off the transformer ratio accordingly.
The transformer ratio also dictates the ratio of the two inductances. That, along with the voltage ratio determine the relative lengths of the charge and discharge times.
All these parameters need to be considered carefully and traded off against each other. One advantage of flyback mode is that you get a wide latitude of tradeoffs.
Of course there are disadvantages to flyback mode too. The magnetic core must be bigger for the same power transfer. In forward mode, ideally the energy is being removed from the magnetic field by the secondary as fast as the primary puts it there. In flyback mode, the entire energy of each pulse is stored in the core at one time.