The solver tolerance factor feature can be used with any integrator and solver type. A smaller value will force more Newton iterations, while a larger value will reduce the number of Newton iterations. This parameter is best adjusted in magnitudes of 10, i.e. 100, 10, 1, 0.1, 0.01 etc. to notice an effect in Newton iteration count.
When would you want to use this feature? It provides a fine level of control over the solver. Ideally you'd never need to adjust the tolerance manually. However, if you're trying to achieve very precise results you may want to reduce the factor.
For example you have some animation, and you want to simulate it as usual, but then also simulate it in reverse, swapping the start and end frames around. Using a quasistatic integrator the start frame sim of the forwards animation should match the end frame sim of the reverse animation solves. It takes an accurate solve to achieve this, hence reducing the solver tolerance factor can be helpful.
Another example where you may desire more accuracy is if you have a simulation with both very large tissues, e.g. 1m in dimensions and very small tissues e.g. 1cm in dimensions. A hypothetical 1mm solver error in vertex position might be visually undetectable in the large tissue, but could stand out in a small tissue. Reducing the solver tolerance factor might help correct this issue if it is due to solver error.
The default solver tolerance used (when solver tolerance factor is 1) is calculated to favour simulation stability and correctness over speed. If you have a simulation using many Newton iterations you could try increasing the solver tolerance factor to reduce the number of Newton iterations and reduce the solve time. If your simulation still looks acceptable then you didn't need the default level of accuracy the solver was calculating your solution to and you can safely benefit from the reduced sim time.
The solver tolerance factor is a fine control of simulation accuracy. If your simulation is experiencing major instability, failed collisions etc. then I would first try increasing the number of substeps used. Substeps is a far more powerful parameter to adjust to improve simulation stability.