"Late game" is part of a suite of concepts that are primarily used in two different kinds of games; multiplayer games with in-match progression (genres like MOBA, RTS, TBS, Grand Strategy), and games where the player progresses in power rather than through a linear story (genres such as MMORPG, ARPG, Lootershooters and many flavors of Roguelike). Games like these tend to have three phases; early, mid and late game. These phases are organic and defined by player consent, change on a per title basis, and are not formalized by the game itself. Games like Mario, DOOM and CS:GO don't have these phases. SWGoH seems to fall into the second category, so we'll focus there.
In this context, late game is when the player has stopped getting anything new, and is instead trying to get better versions of the things they already have. This phase never ends. The player continues to play and accrue power until they decide to stop.
"The next level," as a noun phrase, is a pretty common idiom in US English and is by no means exclusive to games. It is often used in advertising. It usually means a significant, easily noticeable improvement. A new fertilizer may bring a farmer's yield to "the next level." This is the context at work here. "Next level" can also be an adjective phrase. "Next level performance" is especially popular in the automotive advertising industry. "Next level thinking" is meaningless jargon in leadership consulting, but may refer to revolutionary strategies or concepts in other contexts.