The most common and most likely answer to this crossword clue is the 6 letter word LEVELS
I've seen this crossword clue in The Guardian
L
E
V
E
L
S
More clues leading to the result LEVELS
Buddy explains!

The straight part
We know that "flats" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case flats = levels.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does LEVELS mean?
LEVELS noun- A tool for finding whether a surface is level, or for creating a horizontal or vertical line of reference.
- A distance relative to a given reference elevation.
- Degree or amount.
- Achievement or qualification.
- Distance from the root node of a tree structure.
- One of several discrete segments of a game, generally increasing in difficulty and representing different locations in the game world.
- A numeric value that quantifies a character's experience and power.
- A floor of a multi-storey building.
- An area of almost perfectly flat land.
- A school grade or year.
- To adjust so as to make as flat or perpendicular to the ground as possible.
- To destroy by reducing to ground level; to raze.
- To progress to the next level.
- To aim or direct (a weapon, a stare, an accusation, etc).
- To direct or impose (a penalty, fine, etc) at or upon (someone).
- To make the score of a game equal.
- To bring to a common level or plane, in respect of rank, condition, character, privilege, etc.
- To adjust or adapt to a certain level.
- (usually with "with") To speak honestly and openly with.