Microprocessor 8085 regains control of the bus
Your assumption is very spot on. Both the HLDA and HOLD signals are part of the DMA (data memory access), which is able to write directly to memory without having the processor to get involved.



(I am unfortunately not able to track down the original source of this picture as there are many websites with this exact same diagram.)

The DMA transfer happens with these steps:


Signal from device queuing initiation for the DMA by requesting the processor.
The processor will complete its current cycle and will approve the request for the device.
The device then sends an acknowledgement basically saying, "Thank you! We will proceed now..."
The processor senses the acknowledgement and starts monitoring the DMA and makes sure nothing weird happens between the device and memory.
Once the process is finished between the device and memory, the device will tell the processor, "Hey, we're finished here!"
The processor will then receive this message and then it will proceed to whatever it was doing before this all started.


Basically what the HOLD does is that it prevent the processor from doing anything when the DMA process is taking place. Once the device tells the processor that it's all finished, the HOLD will be sent LOW to allow the processor to resume its activities.