My understanding of “Schar Mitzvos b’hai alma leika” based on the way I learned about it many years ago is along the lines of what Sam2 wrote.
Basically, the real schar is in olam haba, but at the same time we see that there is a concept of schar in this world (as seen in Krias Shema “V’nasati matar arstzichem b’ito”) which has to be explained in light of “schar Mitzvos b’hai alma leika”.
I think the explanation I learned was that if you do Mitzvos, Hashem might give you things to make it easier for you to continue doing Mitzvos. Like if He sees You are using your money for worthwhile purposes, He may give you more money. So it’s not really your schar per se’, it’s to help you to continue doing Mitzvos.
Sam2’s explanation sounds like a good explanation as well, and it fits in with the above. It’s a somewhat similar idea.
I have also learned that one of the reasons for “tzadik v’ra lo, rasha v’tov lo” (Righteous people sometimes suffer in this world and wicked people sometimes have it good) is that Hashem wants to give the wicked person his reward in this world so that he won’t get a reward in the World – to- come, whereas He “punishes” the righteous people in this world so that they won’t have to be punished in the World-to-come.
So it would seem that there is a concept of punishment in this world. However, it is for our good and is for the purpose of “cleansing” our Neshamas so that we won’t have to be punished in the next world.
For that matter, the punishment in gehinom is also for our good and for the purpose of cleansing our souls, but it is much better to receive the punishment here since any suffering in this world can not begin to compare to the suffering in Gehinnom.
Also, the purpose of suffering in this world (punishment or not) is always to help you to be a better person while you are still in this world and have a chance to grow. Even if you don’t feel like it’s bringing you closer to Hashem, and even if you feel like it is bringing you further from Hashem, it always has a positive impact on your neshama.