If you inject a current into a model cell with no active components (no voltage-gated channels), you'll see a capacitive response, because that's basically all you have: the membrane acts like a capacitor in parallel with a resistor. Adding a current charges the capacitor and you have an "exponential decay"-shaped change towards a new equilibrium potential.
When you stop injecting current, you'll again have an "exponential decay"-style return to the baseline.
However, the HH model includes active conductances: voltage-gated channels that change state according to voltage. "Action potential" is the name given to a positive-feedback opening of sodium channels: voltages more positive than rest open some channels, which open additional channels, etc. "Threshold" refers to a voltage that triggers a full positive-feedback activation of these channels. Sometimes it's stated to be a particular voltage, but that's not really true; threshold depends on the whole state of the system: voltage and state of all the gates of different channel types.
However, any depolarization will open some voltage-gated channels beyond those open at rest, it's just that up until threshold there aren't enough open to trigger the positive feedback. The reason is that voltage-gated sodium channels don't just open and close, they also inactivate. In the HH model, this is represented by the "h" gate.
If you inject a brief positive current, you'll open some channels and get a little "bump" in the voltage afterwards that then returns to baseline as those channels close/inactivate. If you inject a longer current, you'll open a few more channels and also the "bump" from the open channels will sum with your injected current. If you're right around threshold, that could cause you to exceed the threshold for positive feedback and therefore an action potential.
Though the question is different, a lot of this same reasoning is found in my answer to another question here: Hodgkin huxley neuron not spiking consistently for currents greater than threshold?