What does "lately" mean? And should I always use Perfect times if I say "lately" like this: "I have been playing a game a lot lately." Why is "I was playing a game a lot lately" incorrect?
Lately means 'recently', including up until the present time.
The inclusion of the current time is why 'was' doesn't work, because 'was' refers to a distinct period which ended at some point in the past. 'Have been' covers the same sense of period as the word 'lately'.
in the recent period of time
Source: merriam-webster.com definition of “lately” (emphasis mine)
The word “lately” refers to the recent past (prior to now), not merely the time shortly before a given time.
Think of it as similar to “yesterday” insofar as it is a fixed period behind the present. You wouldn’t say “Some time ago I played a game yesterday.”