Left side; battery then battery connections.
The 10k resistor across the battery is to pull that wire to zero volts when the battery is missing. (important)
Diode, and capacitor.
Next 10k resistor goes to the input of the micro. The resistor is needed because the micro should not look at a voltage higher than its power supply and the battery will be 0.7 volts higher.
Right side; Resistor is the heating element in the vest. MOSFET turns the current on/off. (just saw a mistake. The power to the heating element should come from the battery not through the diode. )
I don't know the Attiny85 very well.
Many of the small micros can use an internal OSC so you don't need an crystal.
Many of the small micros can be run very slow. To reduce power. The 2mA in the above post probably could be cut to 0.5mA or much smaller by cutting the speed. I have run some of the PIC micros at 32khz to reduce power to uAs.
The PWM can be very slow for "heating". If the PWM ran at 1hz it would be fine.
Many of the micros can be put into sleep mode until the timer runs out. Run through your program one and set the timer, then go to sleep. Wake up and loop again.
Shut down pieces of the micro you are not using. (ADC and UART)