The spark gap is a simple, low-tech way of generating high frequencies. It was the only way of generating them before valves and transistors were invented.
As C1 charges up through R1, eventually the air in the spark gap breaks down, becoming a short circuit.
The breakdown of a spark gap happens over just a few nanoseconds. (It turns off much more slowly when voltages on C1 and C2 become equal).
The current through the spark gap is a sort of square up, exponential down. The leading edge contains frequency components up to at least 100 MHz. The bandwidth of a step change is roughly 0.35 / the rise time.
These high frequency components are what excite the tuned circuit, or if you like, are filtered by the tuned circuit, so that only the kHz or MHz component passes through the circuit.
The frequency might have been 500 kHz (Titanic) or later a few MHz. If the loaded Q of the LC circuit is 50, at the frequency of interest, the resonant circuit will "Ring" for 50 cycles, decaying to about half its initial amplitude. This is called a damped wave. At each spark the transmitter sends one of these short, wide, damped RF pulses. (I think as they often had resonant antennas, the loaded Q would have been much lower). The decay process takes about 100 microseconds.
Note also that a Q of 50 will lead to a transmitted signal bandwidth of perhaps 10 kHz, much more depending on what threshold you use for bandwidth, which is quite wide by today's standards and much more than required for morse code. This is why Damped Wave emissions are not allowed by modern radio amateur laws.
Spark gaps up to 1910 simply triggered themselves again and again as C1 charged up enough to cause breakdown. This lead to a random pattern of the bursts of RF, which made a hissing sound in the headphones when detected by a simple AM receiver like a crystal set. You probably know the hissing sound of a randomly triggered spark.
The Titanic was one of the first ships to have a new rotary spark gap, triggered by a rotating disc with metal studs, between the terminals of the spark gap. This made the breakdown happen at regular intervals, a few hundred times per second. These make a new buzzing sound in the receiving headphones.
Lots more about the radios on Titanic, here.