It is called a blocking oscillator. When power is applied, Q1 turns on slightly because it is biased via the LDR and the 50K pot. When Q1 turns on it causes a current to flow in the transformer. This causes the voltage on terminal 39 of the transformer to go more negative turning the transistor on more via C1. This continues until the transformer core saturates. Once it saturates, the magnetic field collapses, driving terminal 39 of the transformer positive and thus turning off Q1. Once the field fully collapses, Q1 will be turned on again and the cycle repeats.
Your diagram didn't need to be that big. Here is a scaled down version (36K vs 1590K) :