As it is mentioned on the history page, nowadays the term Low Level Format
is used to designate a process of zero-filling a storage device.
SSD and zero-filling
Historical meaning of the term Low Level Format doesn't apply to memory chips (and hence to an SSD) at all.
That is because in magnetic disks (and tapes) Low Level Format is used to place discrete marks onto the analog surface.
There are no analog elements in memory chips.
Zero-filling an SSD which is not TRIM-capable is a trick used to improve write performance.
SSD can write data quickly only to a blank (zero-filled) block.
But if there are no blank blocks, then it is necessary to erase some blocks before new data can be written.
The process of erasing takes a long time.
The need to erase blocks decreases performance of an SSD without TRIM because of such an SSD cannot erase blocks in advance.
Therefore, zero-filling applied to an SSD without TRIM speeds write operations up for a while.
Reallocation and zero-filling
At the factory some sectors are reserved to a special pool to compensate for possible surface defects, both manufacturing and operational.
If a sector becomes defective for whatever reason, one of the reserved sectors will be used instead.
This is called "reallocation". Bad sector is a sector on which write operation fails.
Since Low Level Format requires that each sector should be overwritten with zeroes,
it helps detect bad sectors and therefore forces a hard disk to reallocate them.
It is needed to use Low Level Format for reallocation if you are going to work with a second-hand hard drive.
In this case, in addition to removing other people's data, you force a hard disk to reallocate bad sectors, if there are any.