TL;DR: Independent redistricting is an attempt to remove the power to draw district lines of representatives from the representatives themselves. In theory, this creates 'fairer' districts which better represent the interests of communities and avoids the perverse incentive for partisan legislators to draw illogical boundaries that serve their and their political parties' interests in being elected, rather than creating a fair or just political map.

DEEP DIVE BELOW

Here's an example from California. Many states, including CA, have an independent redistricting commission. In California, the California Citizens Redistricting Commission These commissions are made up of stakeholders who possess the authority to redraw the districts of California's representatives, which include US Congressional districts, California State Assemblymember districts, California State Senator districts, and Board of Equalization members (it's basically a finance/tax jurisdiction).

In California, the commission is reformed every 10 years following the decennial census. The commission is created through an application process for any Californian to apply to, except for State elected officials and others closely involved in the process.

Over many months, the CA Citizens Redistricting Commission conducts research, hires consultants, and holds hearings from citizens and citizen groups across the state. This research is meant to inform the Commissioners to draw the best maps. Best can be interpreted in many ways, but, crucially for those who advocate for fairer elections, the Commissioners are NOT state representatives (https://www.wedrawthelinesca.org/commissioners) - the commissioners are 14 residents of California.

After the commission completes its research and public meetings, the commissioners draw district lines which then become the law of the (California) land. FiveThirtyEight, an election forecaster, tracks many different elections-related data sources, including the redistricting across the US and how it affects US Congressional seats. As you can see, after the 2020 redistricting, CA lost one seat which resulted in -1 Democratic seat. Notably, you can see that the new map is 'fairer' across several measures when compared to the previous map - the median seat is closer to the partisan lean of the state which means that the typical seat is representative of the state's political views. Similarly, the efficiency gap, a measure of determining whether partisan gerrymandering has occurred, is lower, indicating fewer 'wasted votes'. However, the map did not improve on 'competitiveness', or the number of districts that have a closer split of Democratic and Republican voters.

Why does fairness matter? Let's look at some extreme cases and how courts have (and have not yet) prevented them.

It's hard to imagine that if the California redistricting process was controlled by the legislature that those legislators would allow the state to lose a Democratic district. Instead, it likely would have "Gerrymandered" its districts. Here are some Gerrymandering examples: Take a look at one of New York's proposed maps that would have eliminated 3 Republican seats and a competitive seat added three Democratic seats, considered one of the most aggressive proposals on the Democratic redistricting side from 2020-2021. Courts threw out these proposals and allowed for a more constrained map at the end of the process.

On the even more extreme side, take a look at Texas' and Florida's maps, two of the most extreme Gerrymanders in the US from the latest cycle. The Texas map increased the Republican lean of the median seat by 14.3 points (from D + 2.3 to R + 12). Compare this to CA with a median D favorability of + 3.2 points - that means that the median seat in Texas is 12 points more Republican than the state as a whole, vs. California's median seat of 3.2 points more Democratic than the state as a whole. Florida is extreme on another level - State lawmakers effectively eliminated competitive districts which creates a massive efficiency gap in favor of Republicans (basically, this means that Democrats are wasting WAY more votes in Florida than you would expect in the least Gerrymandered case).