The cogs on the transmission interlock with each other with big pins on their sides that slot into oval holes on the mating cog. The shift forks move the cogs back and forth on the spindle, interlocking the different gears together resulting in a different overall ratio coming out the countershaft end relative to the RPM of the engine.
So, if the pins and holes on the mating cogs are not aligned, the gear will move a bit but not all the way and won't disengage from the one it was already mated to (i.e. stuck in 2nd).
Pulling in the clutch again / letting the transmission rotate a bit more ultimately will end up with the gears aligned and let the cog move over and mate with the new one to move into 3rd.
Easiest to understand if you actually look at it happening in person or on YouTube.
If your clutch isn't fully disengaging, allowing the transmission to slow down/stop, this could hinder you as well, but as f22hb says, you can shift gears without the clutch if you match RPM to road speed well enough so the gears in the tranny are basically rotating at the same speed allowing the pins/holes to line up and not clash with each other as you shift.