What is the purpose of higher-resistance resistors in Astable Multivibrator circuit?
The purpose of the Base resistors is to integrate the supply voltage ( when Collector is near 0V and pullup the cap voltage from -Vx to +Vbe to start conduction on the other side then switch ON Q (toggle outputs) and repeat for the other side.

This design works best on low voltage.Here with a variable Battery and reduced collector R to raise LED current at lower voltage. 3 to 5V


When Q1 conducts hard from C2 it  dumps a negative pulse to Vbe2 thru C1 to reverse bias Q2 for a period reduced by T=R1C1 but then ramps up slower by T=R2C1 until Vbe reaches the conduction threshold. 

Due to the current gain each device switches hard then the base current is slowly reduced.  But if the Rb/Rc ratio is too high the LED current will decay quickly and the clock slows down.

The base current is integrated so RbC affects the slew rate and cycle frequency so V+/RbC is the major factor for determining the frequency . However Vbe has a max reverse voltage often =5V and if V+ is > 6V then there is a damage risk to the above design to Vbe reverse limits. So a reverse diode was often added to prevent this failure. But that also reduces the ramp swing and thus raises the frequency.

A better simpler design now uses CMOS Schmitt triggers.
This can work with prudent choices of R=33M and C values with low leakage caps to make cycle times ~ 1 Minute possible or > 50 MHz with 10pF, 1K

Both use the negative feedback of input current to reach a conduction threshold then toggle the integration of voltage to reach threshold to toggle switching again.

note the polarized caps can usually handle up to 10% of its voltage rating in reverse voltage but is never published as such.  Here they see the Vbe voltage in reverse across the cap for a half cycle.  

Vbe however must never be reverse biased below -5V as indicated as Vr MAX in all datasheets.