It does appear to be a silly circuit for a few reasons.
It's supposed to be switching on/off yet it has bias resistors for Q1, and a coupling capacitor C7. That's what you'd have in a linear amplifier, it's confusing.
I think C7 is there to ensure Q1 is not always on, which would roast the speaker. The 555 output might be high at idle? Have to see the rest of the 555 circuit to see why C7 is even needed. C7 limits the maximum on-time pulse width for Q1.
But because the 555 drive goes through big C7, and the E-B junction of Q1 is like a diode, the capacitor will eventually charge up with net DC. It also looks like the wrong polarity (- to 555 timer out). I don't know the frequency and duty cycle here.
When the 555 timer output is +ve, C7 charges through the E-B junction and R3. This turns on Q1 and the base pulses up to around +1V.
When the 555 timer output is GND, C7 discharges through R2 and R3 and takes Q2 base negative to -4.5V.
OP I think the circuit is not so great and could be done differently, depending on the rest of the 555's use.