The way I understand it is that, without the Trim command, the SSD doesn't know that a given sector on the drive has good or stale data on it. You delete a file, the blocks that contained it's data become available to the OS/filesystem for reuse. Until that logical block is written to again (with new data), it assumes that it is still good, so when space is needed at the lower level, it can copy the data to a new area when it needs to make space for an erase. Or moved around for wear leveling (which essentially means that an unchanged file written to the SSD years ago, and never logically moved by a defragmentation, will still move to different physical sectors on occasion).
TRIM just allows the OS to tell the drive that a block is free without waiting.
Without trim, eventually the drive will probably think the whole partition is in use, as data is eventually written over the whole partition. Then, it has to use garbage collection to manage smaller blocks against the overprovisioning space on the drive that isn't logically visible to the OS/filesystem, but only at the drive level. So writes are always possible even it the drive is "full".